Download Fundamentals of Probability Outline Introduction

Outline
Part One
I. Random Variables and Their Probability Distributions
II. Joint Distribution, Conditional Distributions, and Independence.
III. Features of Probability Distributions
Fundamentals of Probability
Read Wooldridge, Appendix B
Part Two
IV. Features of Joint and Conditional Distributions
V. The Normal and Related Distributions I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Fundamentals of Probability: Part One . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
I. Random Variables and Their Probability Distributions
Introduction
• Strategy 2:
Should the airline accept more than 100 reservations?
• Probability is of interest for students in business, economics, and other fields.
– This policy runs the risk of the airline having to compensate people who are bumped from the overbooked flight.
Example: Airline is trying to maximize profits from the available 100 seats. • Strategy 1: Should the airline accept reservations at most 100 seats?
• Can we decide on the optimal or the best number of reservations the airline should make?
• There is a chance that those who book might not show up. This results in lost revenue to the airline
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
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• We can use basic probability to arrive at a solution.
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I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
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Random Variables
Experiment
• Experiment: flip a coin ten times and count the number of heads.
• Definition
A random variable is one that takes on numerical values and has an outcome that is determined by an experiment.
• Definition:
An experiment is the procedure that can be repeated and has a well‐defined set of outcomes.
•
– Carrying out the coin‐flipping procedure again and again
– Number of heads appearing in an integer from 0 to 10.
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Random Variables and Outcome
A. Discrete
B. Continuous
A. Discrete
B. Continuous
Example: Income of 20 randomly chosen households in the United States.
• Notations
Random Variables – are denoted by an uppercase letters; eg, X,Y
Particular Outcomes – are denoted by a smallcase
letters; eg, x,y.
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
• Subscripts: we indicate large collections of random variables by using subscripts.
– particular outcome
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
Random Variables and Outcome
X =“number of heads” appearing in
– random variable
10 flips of a coin
I. Random Variables and Their Probability Distributions
Example: a random number is the number of heads appearing in an experiment (10 flips of a coin).
I. Random Variables and Their Probability Distributions
Example: Note that the random variable X takes on a value in a set {0, 1, 2, …, 10} x=6
A. Discrete
B. Continuous
– Random variables: X1, X2, … X20
– Particular outcomes: x1, x2, … x20
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I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
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Random Variables and Outcome
A. Discrete
B. Continuous
Bernoulli or Binary Random Variable
Example: Consider tossing a single coin
1) What are the outcomes?
heads and tails
• Definition:
A random variable that can only take on the values zero or one (0 or 1) is called a Bernoulli (or binary) random variables.
event X=1
success
event X=0 failure
2) This is an example of qualitative events.
3) Define the random variable as follows.
X=1 if the coin turns up heads
X=0 if the coin turns up tails. I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
•
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Discrete Random Variable
A. Discrete
B. Continuous
Definition:
A discrete random variable is one that takes on only a finite or countable infinite number of values.
– we will concentrate on discrete random variables that takes on only a finite number of values. I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
A. Discrete
B. Continuous
•
Any discrete random variable is completely described by listing its possible values and associated probabilities that it takes on each value.
•
A Bernoulli random variable is the simplest example of a discrete random variable. Example: coin flipping
P(X=1) = ½ “the probability that X equal one is one‐half”
P(X=0) = ½ For our purpose,
I. Random Variables and Their Probability Distributions
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
Discrete Random Variable and Probability Distribution
• “Countable infinite” means even though an infinite number of values can be taken as a random variable, those values can be put in one‐to‐one correspondence with the positive integers.
•
A. Discrete
B. Continuous
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Discrete Random Variable
•
A. Discrete
B. Continuous
Discrete Random Variable and Probability Distribution
Example: airline example.
Suppose X takes on k possible values {x1, x2,,…xk} and their associated probabilities {p1, p2,….,pk}, there are two properties.
X=1 X=0
(1) P(X=xj) = pj
P(X=1) = 
P(X=0) = 1‐
(2) p1+ p2 + … + pk =1
The sum of all probabilities is equal to one (1).
•
•
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Discrete Random Variable and Probability Distribution
A. Discrete
B. Continuous
Special Notation: X = Bernoulli ().
It is read as “X has a Bernoulli distribution with probability of success equal to .” I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
A. Discrete
B. Continuous
Example: X is the number of free throws made out of two attempts.
The probability density function (pdf) of X summarizes the information concerning the possible outcomes of X and the corresponding probabilities:
f(xj) = pj ; j = 1, …, k
Notes 1) f(x)=0 for any x not equal to xj for j
2) pdf provides information of the likely outcomes of the random variable.
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
(Interpret! when  equals 0.75)
Discrete Random Variable and Probability Distribution
• Definition: pdf of X
•
if the person shows up for the reservation.
otherwise
– To completely describe the behavior of X, define  as the probability that the customer shows up after making a reservation. “The probability that X takes on the value xj is equal to pj.” I. Random Variables and Their Probability Distributions
A. Discrete
B. Continuous
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
•
X can take on three values, {0, 1, 2}
•
pdf is given by
f(0) = 0.2; f(1) =0.44; f(2)=0.36. (see graph)
•
What is the probability that the player makes at least one free throw.
P(X  1) = P(X=1) + P(X=2) 0.44+0.36 = 0.80
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Discrete Random Variable
A. Discrete
B. Continuous
Continuous Random Variable
•
A. Discrete
B. Continuous
Definition
A variable X is a continuous random variable if it takes on any real value with probability zero (0). I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
Continuous Random Variable
•
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
A. Discrete
B. Continuous
•
“A function that can be graphed without lifting your pencil from the paper.”
•
Idea: a continuous random variable takes on so many possible values.
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
Continuous Random Variable
•
Probability density function for continuous rv.
P(a<X<b) (Draw graph!)
Cumulative Distribution Function (cdf)
•
For discrete rv, cdf is obtained by summing the pdf over all values xj such that x<xj.
•
For continuous rv, cdf is equal to the area under the pdf [f(x)] to the left of xj.
•
This is the integral of the function f
between points a and b.
A cdf is an increasing (or at least nondecreasing) function of x.
Given x1<x2, then P(X  x1) < P(X  x2); i.e., F(x1)<F(x2).
The area under
the pdf
between points
a and b.
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
A. Discrete
B. Continuous
If X is a random variable, then its cdf is defined for any value of x by
F(x) = P(X  x)
– For positive probabilities, a continuous rv X lies between certain values (a and b); ie.,
I. Random Variables and Their Probability Distributions
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Continuous Random Variable
A. Discrete
B. Continuous
Problem B.1
Properties:
1) For any number c, P(X>c) = 1‐F(c)
B.1 Suppose that a high school student is preparing to take the SAT exam. – Explain why his or her eventual SAT score is properly viewed as a random variable.
[ans.]
2) For any numbers a<b, P(a  X b) = F(b) ‐ F(a)
•
A. Discrete
B. Continuous
For econometrics, we will use cdf to compute probabilities only for continuous random variables; eg, normal distribution. P(X ≥ c) = P(X > c)
P(a  X b) = P(a < X b) = P(a  X < b) = P(a < X < b)
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
Problem B.1
Problem B.4
Before the student takes the SAT exam, we do not know – nor can we predict with certainty – what the score will be. •
The eventual SAT score clearly satisfies the requirements of a random variable. •
A. Discrete
B. Continuous
B.4 For a randomly selected county in the United States, let X represent the proportion of adults over age 65 who are employed, or the elderly employment rate. Then, X is restricted to a value between zero and one. Suppose that the cumulative distribution function for X is given by B.1
•
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
I. Random Variables and Their Probability Distributions
F(x) = 3x2 – 2x3
The actual score depends on numerous factors, many of which we cannot even list, let alone know ahead of time. (The student’s innate ability, how the student feels on exam day, and which particular questions were asked, are just a few.) for 0x 1. Find the probability that the elderly employment rate is at least 0.6 (60%). [ans.]
ACK
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
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I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Problem B.4
II. Joint Distribution, Conditional Distributions, and Independence
B.4
X: proportion of adults over age 65 who are employed
Find the probability that the elderly employment rate at least 0.6 •
•
We want P(X .6). •
Because X is continuous, this is the same as P(X > .6) = 1 – P(X  .6)
= 1 ‐ F(.6)
F(.6) = 3(.6)2 – 2(.6)3 = .648. = 1 ‐ .648
= .352
A. Joint Distrib.
B. Conditional
In business and economics, we may be interested in the occurrence of events of more than one random variables.
Example: •
– The airline may be interested in the probability that a person makes a reservation and is a business traveler. This is an example of joint probability.
– The airline may be interested in the probability that a person makes a reservation, conditional that he is a business traveler. This is an example of conditional probability.
One way to interpret this is that 35% of all counties have an elderly employment rate of .6 or higher.
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
I. Random Variables and Their Probability Distributions
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Joint Distribution and Independence
•
•
A. Joint Distrib.
B. Conditional
Let X and Y be discrete random variables. Then, (X,Y) have a joint distribution.
Independence
•
Random variables X and Y are said to be independent if and only if
fX,Y(x,y) = fX(x)fY(y)
fX(x) is the pdf of X and fY(y) is the pdf of Y.
Given pdfs of X and Y, it is easy to obtain a joint pdf. I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
II. Joint Distribution, Conditional Distributions, and Independence
A. Joint Distrib.
B. Conditional
•
The joint distribution can be fully described by the joint density function of (X,Y)
fX,Y(x,y) = P(X=x, Y=y) or
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Joint Distribution and Independence
f(x,y) = P(X=x, Y=y)
•
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
II. Joint Distribution, Conditional Distributions, and Independence
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
•
Note that in the context of more than one random variable, – fX(x) and fY(y) are marginal pdfs.
•
Intuitively, independence refers to knowing the outcome of X does not change the probabilities of the possible outcomes of Y and vice versa.
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
II. Joint Distribution, Conditional Distributions, and Independence
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Joint Distribution and Independence
A. Joint Distrib.
B. Conditional
Example B.1: Free Throw Shooting
A. Joint Distrib.
B. Conditional
Example: Free Throw Shooting
•
Given
– Let X and Y be a Bernoulli random variable. Note that a Bernoulli random variable is a zero‐one random variable.
fX,Y(x,y) = fX(x)fY(y)
•
X = 1 Y = 1 It is the same as
he makes the first free throw
he makes the second free throw
– Suppose that she or he is an 80% free‐throw shooter; i.e.,
P(X=x,Y=y) = P(X=x)P(Y=y)
P(X=1) = P(Y=1) = 0.8. Here we are interested in the number of success in a sequence of Bernoulli trials.
That is, the probability that X=x and Y=y is the product of the two probabilities P(X=x) and P(Y=y).
•
What is the probability of the player making both free throws?
P(X=1)P(Y=1) = (0.8)(0.8) =0.64
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
II. Joint Distribution, Conditional Distributions, and Independence
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Example B.1: Free Throw Shooting A. Joint Distrib.
B. Conditional
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
II. Joint Distribution, Conditional Distributions, and Independence
Joint Distribution and Independence
•
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
A. Joint Distrib.
B. Conditional
Given a random variable X and a function g(.), – create g(X) as a new random variable.
Example: continue
•
If the chance of making the second free throw depends on whether the first was make – that is, X and Y are not independent.
•
•
Intuitively, independence refers to knowing the outcome of X does not change the probabilities of the possible outcomes of Y and vice versa.
•
Example
– X is a random variable, so are X2, exp(X) and log(X).
Useful fact:
If X and Y are independent and we define new variables, g(X) and h(Y)
for any function g and h, then
• these two random variables are also independent.
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
II. Joint Distribution, Conditional Distributions, and Independence
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
II. Joint Distribution, Conditional Distributions, and Independence
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Joint Distribution and Independence
A. Joint Distrib.
B. Conditional
Joint Distribution and Independence
Example: Airline
A. Joint Distrib.
B. Conditional
Example: Airline (continue)
– Suppose the airline accepts n reservations. For each i=1, …, n, – Define X be the number of customers showing up out of n reservations
X = Y1 + … + Yn
– let Yi denote the Bernoulli random variable indicating whether the customer shows up: Yi = 1, if he shows up
Yi = 0, otherwise
– Assume Yi has the probability of success  and the Yi are independence. Then, the probability density function of X is – Let  be the probability of success. Each Yi has a Bernoulli () distribution.
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
II. Joint Distribution, Conditional Distributions, and Independence
– When a random X has a pdf as above, we write
X  Binomial (n, ).
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Joint Distribution and Independence
A. Joint Distrib.
B. Conditional
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
II. Joint Distribution, Conditional Distributions, and Independence
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Joint Distribution and Independence
A. Joint Distrib.
B. Conditional
Example: n!
100!
n

1
 
 x  x !(n  x)! 100!
Let n=100,  = 0.95
• What is the probability that none shows up?
P(X=0) = 1*(.95)0*(.05)100 = 1*1*(7.8886E‐131) = 7.8866E‐131
Let n=100,  = 0.5
• What is the probability that at least one person shows up?
P(X1) = 1 – P(X=0) = 1 ‐ (7.8886E‐131) = 1
n!
n
 
 x)!
!(
x
x
n
 
– n! = n(n‐1)(n‐2)   1
– 0! = 1
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
II. Joint Distribution, Conditional Distributions, and Independence
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
II. Joint Distribution, Conditional Distributions, and Independence
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Conditional Distributions
A. Joint Distrib.
B. Conditional
Conditional Distributions
•
•
Generally, Y is related to one or two or more variables. •
How X affects Y is contained in the conditional distribution of Y given X. A. Joint Distrib.
B. Conditional
This information is summarized in the conditional probability density function, defined as
fYX(yx) = fX,Y(x,y)/fX(x)
f(yx) = f(x,y)/f(x)
for all values of x such that fX(x)>0.
•
The conditional pdf when X and Y are discrete is
fYX(yx) = P(Y=yX=x) = P(X=x,Y=y)/P(X=x)
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
II. Joint Distribution, Conditional Distributions, and Independence
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Conditional Distributions
A. Joint Distrib.
B. Conditional
A. Joint Distrib.
B. Conditional
Example: Free Throw Shooting
• Suppose X is the first free throw and Y is the second free throw.
fYX(11) = .85
f(yx) = f(y)
– If the player makes the first free throw, the probability of making the second free throw is 0.85.
f(xy) = f(x)
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
II. Joint Distribution, Conditional Distributions, and Independence
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Example B.2 Free Throw Shooting
• When X and Y are independent, knowledge of the value taken by X tells us nothing about the probability that Y takes on various values, and vice versa. That is,
fYX(yx) = fY(y) (show !)
fXY(xy) = fX(x)
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
II. Joint Distribution, Conditional Distributions, and Independence
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
II. Joint Distribution, Conditional Distributions, and Independence
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
A. Joint Distrib.
B. Conditional
Example B.2 Free Throw Shooting
III. Features of Probability Distributions
A. Central tendency
B. Variability
fYX(11) = .85
•
•
• We are interested in few aspects (concepts) of the distributions of random variables.
Interpret the following:
1) fYX(10) = .70
2) fYX(01) = .15
3) fYX(00) = .30
4) P(X=1) = .80
5) P(X=1,Y=1)
1) Measure of central tendency
2) Measure of variability or spread
3) Measure of association between two random variables.
Can we compute P(X=1,Y=1)?
P(X=1,Y=1) = P(Y=1X=1)P(X=1)
= (0.85)*(0.8) = .65
Why??
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
II. Joint Distribution, Conditional Distributions, and Independence
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
A Measure of Central Tendency
A. Central tendency
B. Variability
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
A Measure of Central Tendency
A. Central tendency
B. Variability
• Suppose X is a random variable. • Expected value
– one of the most important probabilistic concepts – names: expected value, the expectation of X
– notations: E(X), X, 
– The expected value of X is the weighted average of all possible values of X. – The weights are determined by the probability density function. • If X represents some variable in a population,
– The expected value is the population mean.
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III. Features of Probability Distribution
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III. Features of Probability Distribution
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A Measure of Central Tendency
•
A. Central tendency
B. Variability
Example B.3 Computing an Expected Value
Precise (mathematical) definition A. Central tendency
B. Variability
Example:
– X takes on the values ‐1, 0, 2 with probabilities 1/8, ½, and 3/8
Suppose X is a discrete random variable taking on a finite number of values {x1, x2
,..., xk} and f (x) denote the probability function. The expected value of X is
E(X) = ‐1(1/8) + 0(1/2) + 2(3/8) = 5/8
E(X2) Notes:
1) The expected value can be a number that is not even a possible value of X
2) It can be deficient for summarizing the central tendency of certain discrete random variables.
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III. Features of Probability Distribution
A Measure of Central Tendency
•
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
A. Central tendency
B. Variability
• Notes
1) E(X) is always a number that is a possible outcome of X.
2) It is computed using integration.
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A. Central tendency
B. Variability
•
For discrete random variable, the expected value of g(X) is simply a weighted average:
•
For continuous random variable, the expected value of g(X) is
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III. Features of Probability Distribution
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
A Measure of Central Tendency
Suppose X is a continuous random variable, the expected value defined as an integral:
III. Features of Probability Distribution
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III. Features of Probability Distribution
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A. Central tendency
B. Variability
A Measure of Central Tendency
A Measure of Central Tendency
A. Central tendency
B. Variability
Example: •
Let g(X) =X2. •
The random variable X takes on the values ‐1, probabilities 1/8, 0, ½, 2 3/8
with •
Suppose X and Y are two discrete random variables. •
The expected value of g(X,Y), given X and Y taking on values {x1, …, xk} and {y1, …, ym}, respectively, is
k
E(X2) •
•
h 1 j 1
where fX,Y(xh, yj) is the joint probability density function of (X,Y).
Note that E(X2)  [E(X)]2
Example: 13/8  [5/8]2
This suggests that for a nonlinear function, g(X),
E[g(x)]  g[E(x)].
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III. Features of Probability Distribution
•
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Properties of Expected Value A. Central tendency
B. Variability
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Properties of
Expected Value
A. Central tendency
B. Variability
Property E.2
• For any constants a and b, E(aX + b) = aE(X) + b
Property E.1
For any constant c, E(c) = c
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I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
III. Features of Probability Distribution
There are a few simple rules that we need to know to appreciate certain manipulations for important results.
III. Features of Probability Distribution
m
E[ g ( X , Y )]   f X ,Y ( xh , y j ) g ( xh , y j )
= ‐12(1/8) + 02 (1/2) + 22 (3/8) = 13/8
Example: Let E(X) = 
Y = X‐
E(Y) = E(X) –  = 0
Where a = 1 and b = ‐
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III. Features of Probability Distribution
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Properties of Expected Value
A. Central tendency
B. Variability
Properties of Expected Value Property E.3
• If {a1, …, an} are constants and {X1, …, Xn} are random variables, then
E(a1X1 + … + anXn) = a1E(X1) + ... + anE(Xn)
Example: Y is in Fahrenheit and X is in Celsius.
Y = 32 + (9/5)X
•
What is the expected temperature in Fahrenheit if the expected temperature in Bangkok is 25 degree Celsius?
E(X) = 25
E(Y) = 77
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III. Features of Probability Distribution
•
•
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Properties of Expected Value A. Central tendency
B. Variability
Using summation notation,
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III. Features of Probability Distribution
A. Central tendency
B. Variability
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Example B.5 Finding Expected Revenue
A. Central tendency
B. Variability
Example:
Let X1, X2, X3 are the numbers of small, medium, and large pizzas with E(X1)=25, E(X2)=57, E(X3)=40
If ai=1, then
The prices of small, medium, and large pizzas are $5.5, $7.6, an d $9.15
1)The expected revenue from pizza sales is
E(5.5X1 +7.6X2 + 9.15X3 ) = 5.5E(X1) + 7.6E(X2) + 9.15E(X3)
=5.5(25) + 7.6(57) + 9.15(40) =137.5+433.2+366 = 936.70
Read: The expected value of the sum is the sum of expected value.
• This property is often used!
2) The actual revenue on any particular day will generally differ from this value
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III. Features of Probability Distribution
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Example B.5 Finding Expected Revenue
•
A. Central tendency
B. Variability
Summary: Expectation and Summation
Expected Value
Given X  Binomial(n,). Then, E(X) = n
Summation
Show:
E.1 E(c) = c
s.1 1) Note that X = Y1+ …+Yn, where Yi  Bernoulli ().
E(X) = E(Y1+ …+Yn) = E(Y1) + …+ E(Yn) = n
E.2 E(aX+b) = aE(X) + b
s.2 – The expected number of successes in n Bernoulli trials is the number of trials times the probability of success on any particular trial.
2) Suppose that  =.85. What is the expected value of people showing up when there are 120 bookings?
E(X) = 102
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
The Median A median is another measure of central tendency. It is sometimes denoted as med(X).
•
If X is discrete and takes on a finite number of values, the median is obtained by ordering the possible values of X and select the value in the middle.
∑
If ai=1, E ∑
∑
s.3
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III. Features of Probability Distribution
A. Central tendency
B. Variability
•
E.3 E ∑
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
The Median
A. Central tendency
B. Variability
Example: X takes on the values
{‐4, 0, 2, 8, 10, 13, 17}
• The median is 8
Example: {‐5, 3, 9, 17}
• What is the median? (6)
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III. Features of Probability Distribution
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III. Features of Probability Distribution
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A. Central tendency
B. Variability
The Median
•
Measures of Variability
A. Central tendency
B. Variability
What is one special case that mean and median are equal?
• Expected value – a measure of central tendency Mathematically, the condition is
f( + x) = f( - x)
• Variance – a measure of dispersion
– Variance shows how the individual x values are spread or distributed
about its mean value.
Symmetric distribution
Ans. This could
occur if the probability
distribution of X is
symmetrically
distributed about the
value µ.
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III. Features of Probability Distribution
Variance
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I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
A. Central tendency
B. Variability
A. Central tendency
B. Variability
Variance
Suppose there are two pdfs with the same mean. If the distribution of X is more tightly centered about the mean than is the distribution of Y, – we say that the variance of X is smaller than that of Y.
•
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
III. Features of Probability Distribution
Given a random variable X and the population mean =E(X).
1) The expectation of X measures the central tendency
2) The variance, a measure of variability, measures how far X from , on average.
•
Variance tells us the expected distance from X to its mean:
Var(X) = E(X‐)2
1) We works with squared difference. 2) The squaring serves to eliminate sign from the distance measure.
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III. Features of Probability Distribution
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A. Central tendency
B. Variability
Variance
Variance and Bernoulli
Example: [Binomial]
• Notations for Variance
2x, or simply 2, when the context is clear.
– Given Y  Bernoulli () then, E(Y) = 
– Since Y2=Y (say Y=1 if the person shows up for reservation) and E(Y2) = ; thus,
• Show that 2 = E(X2) – 2
2
Var(Y) =  – 2 = (1‐ )
= E(X2 ‐2X – 2) = E(X2) ‐2E(X) – 2
= E(X2) – 2
• Note that Var(Y) = E(Y2) – 2
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Variance and Properties
A. Central tendency
B. Variability
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III. Features of Probability Distribution
A. Central tendency
B. Variability
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Variance and Properties
A. Central tendency
B. Variability
Property V.2
Property V.1
• Var(X)=0 if and only if there is a constant c such that P(X=c)=1, in which case, E(X)=c
•
For any constants, a and b, Var(aX+b) = a2Var(X)
Notes:
1) Adding a constant to a random variable does not change the variance.
2) Multiplying a random variable by a constant increases the variance by a factor equal to the square of that constant.
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III. Features of Probability Distribution
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Variance and Properties
A. Central tendency
B. Variability
Standard Deviation
Example:
• What is the variance of Y when Y = 32 + (9/5)X
•
Var(Y) = (9/5)2Var(X)
= (81/25)Var(X)
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Standard Deviation
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
The standard deviation of a random variable, sd(X), is simply the positive square root of the variance:
sd(X) = +[VAR(X)] ½
•
Notations for he standard deviation or simply x

when the random variable is understood.
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Standard Deviation
A. Central tendency
B. Variability
Property SD.2
• For any constants, a and b, sd(aX+b) = asd(X)
Property SD.1
• For any constant c, sd(c)=0
III. Features of Probability Distribution
•
III. Features of Probability Distribution
A. Central tendency
B. Variability
A. Central tendency
B. Variability
71
•
If a>0, then
sd(aX+b) = asd(X)
•
Property SD.2 makes standard deviation more natural to work with than variance.
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Standard Deviation
A. Central tendency
B. Variability
Standardizing a Random Variable
Example:
•
• Let X be a random variable. A new random variable can be obtained by Let X be a random variable in thousand of dollars
Let Y=1000X; i.e., Y is a random variable in dollars
Suppose E(X) = 20 and sd(X) = 6, What are E(Y), sd(Y), and Var(Y)?
– subtracting off its mean  and – dividing by its standard deviation :
•
Ans.
E(Y) = 20,000 and sd(Y) = 6,000
Var(Y) = (1000)2Var(X) = 1,000,000Var(X); •
Note that the variance of Y is one million times larger than the variance of X.
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Standardizing a Random Variable
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III. Features of Probability Distribution
A. Central tendency
B. Variability
This procedure is known as standardizing the random variable X. Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
A standardized random variable Z has a mean of zero and a variance equal to one.
where a = 1/ and b = ‐ /
•
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
A. Central tendency
B. Variability
Z = aX+b
– Z is called standardized random variable. – In introductory statistic courses, it is called Z‐transform of X.
III. Features of Probability Distribution
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Standardizing a Random Variable
•
•
A. Central tendency
B. Variability
75
Z has zero Mean
E(Z) = E(X/ – /) = (1/)E(X) –(/) Note that E(X) = 
= 0
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Standardizing a Random Variable
A. Central tendency
B. Variability
Standardizing a Random Variable
Example: Let E(X)=2 and Var(X)=9
• What is the standardized random variable? What are its mean and variance?
Z = aX+b
where a = 1/ and b = ‐/
•
A. Central tendency
B. Variability
•
Z has unit variance
Var(Z) = Var(X/ – /)
Note that Var(X) = 2
= (1/2)Var(X)
= 1
Ans.
1) Standardized random variable:
Z = (X‐)/ = (x‐2)/3
2) Z has expected value zero.
3) Z has variance one.
.
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III. Features of Probability Distribution
Variance: Var(X) = E(X‐)2
Standard Deviation: s.d(X) = E.1 E(c) = c
v.1
Var(c) = 0
s.d.1 s.d.(c) = 0
E.2 E(aX+b) = aE(X) + b
v.2
Var(aX+b) = a2Var(X)
s.d.2 s.d.(aX+b) =as.d.(X)
E.3 E ∑
∑
If ai=1, E ∑
∑
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Problem B.7
Summary: Expectation and Summation
Expected Value: E(X)
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III. Features of Probability Distribution
A. Central tendency
B. Variability
B.7 If a basketball player is a 74% free‐throw shooter, then, on average, how many free throws will he or she make in a game with eight free‐throw attempts? [ans.]
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III. Features of Probability Distribution
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A. Central tendency
B. Variability
Problem B.7
Problem B.8
B.7
• In eight attempts the expected number of free throws is 8(.74) = 5.92, or about six free throws.
• B.8 Suppose that a college student is taking three courses: a two‐credit course, a three credit course, and a four‐credit course. The expected grade in the two‐
credit course is 3.5, while the expected grade in the three‐ and four‐credit courses is 3.0. What is the expected overall grade point average for the semester? [ans.]
(Remember that each course grade is weighted by its share of the total number of units.)
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III. Features of Probability Distribution
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
III. Features of Probability Distribution
A. Central tendency
B. Variability
Problem B.8
Problem B.9
B.8
The weights for the two‐, three‐, and four‐credit courses are 2/9, 3/9, and 4/9, respectively. • B.9 Let X denote the annual salary of university professors in the United States, measured in thousands of dollars. Suppose that the average salary is 52.3, with a standard deviation of 14.6. Find the mean and standard deviation when salary is measured in dollars. [ans.]
• Let Yj be the grade in the jth course, j = 1, 2, and 3, let X be the overall grade point average. Then X = (2/9)Y1 + (3/9)Y2 + (4/9)Y3
• The expected value is E(X) = (2/9)E(Y1) + (3/9)E(Y2) + (4/9)E(Y3)
= (2/9)(3.5) + (3/9)(3.0) + (4/9)(3.0)
= (7 + 9 + 12)/9  3.11.
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III. Features of Probability Distribution
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
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III. Features of Probability Distribution
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A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Problem B.9
IV. Features of Joint and Conditional Distributions
B.9
• If Y is salary in dollars then Y = 1000X, and so the expected value of Y is 1,000 times the expected value of X, and the standard deviation of Y is 1,000 times the standard deviation of X. • Therefore, the expected value and standard deviation of salary, measured in dollars, are $52,300 and $14,600, respectively.
• The joint probability density function (pdf) of two random variables completely described relationship between them.
• There are two summary measures that describe how, on average, X and Y vary with one another: covariance and correlation.
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III. Features of Probability Distribution
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IV. Features of Joint and Conditional Distributions
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Covariance 86
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Covariance
Let X=E(X) and Y=E(Y) and consider the variable (X‐ X)(Y‐ Y).
• The covariance between two random variables, X and Y, measures linear dependence between two variables. It is defined as the expected value of the product (X‐X) and (Y‐
Y):
– X > X implies “X is above its mean”
– Y > Y implies “Y is above its mean”
– If X > X and Y > Y, then (X‐ X)(Y‐ Y) > 0
– If X < X and Y < Y, then (X‐ X)(Y‐ Y) > 0
– If X > X and Y < Y, then (X‐ X)(Y‐ Y) < 0
Cov(X,Y) = E(X‐X)(Y‐ Y)
• which is sometimes denoted as XY.
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IV. Features of Joint and Conditional Distributions
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A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Covariance
•
Covariance and Properties
It can be shown that Property COV.1 [Binomial]
• If X and Y are independent, then Cov(X,Y) = 0.
Cov(X,Y) = E(XY) ‐ XY
•
Show
•
Cov(X,Y) •
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Cov(X,Y) = E[(X ‐ X)(Y ‐ Y)]
= E[(X ‐ X)Y]
= E[X(Y‐ Y)]
= E(XY) ‐ XY
Note that E(XY)=E(X)E(Y)] = XY
•
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Covariance and Properties
= E(XY) ‐ XY
= E(X)E(Y) ‐ XY = 0
If E(X)=E(Y)=0, then Cov(X,Y) = E(XY).
IV. Features of Joint and Conditional Distributions
Show The converse is not true. I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Covariance and Properties
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
The converse is not true! If the covariance of X and Y is zero, this does not imply that X and Y are independent.
Property COV.2
• If a1, a2, b1, and b2 are constants, then
Cov(a1X+b1,a2Y+b2) = a1a2Cov(X,Y)
Example: (See CE.5)
• There are random variables X such that if Y=X2, then Cov(X,Y) = 0 – [Any random variable with E(X)=0 and E(X3)=0 has this property]
– Show Cov(X,Y) = 0.
Cov(X,Y) = E(XY) ‐ XY = E(X3) – 0*Y = 0 •
• Covariance can be altered by multiplying by a constant
If Y=X2, then X and X2 are clearly not independent: once we know X, we know Y.
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IV. Features of Joint and Conditional Distributions
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I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
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Correlation Coefficient •
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Correlation Coefficient
Suppose we want to find the relationship between amount of earnings X and annual earnings Y. – We can find the covariance between X and Y, – but it depends on the units of measurement. •
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
•
The deficiency of the covariance measure is overcome by the correlation coefficient of X and Y:
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
XY = XY
XX 2) It sometimes called the population correlation.
3) The sign depends on XY. 4) The magnitude of Corr(X,Y) is easier to interpret and invariant to the units of measurement (see to Property Corr.1).
5) If X and Y are independent, then Corr(X,Y) = 0.
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Correlation Coefficient
Notes
1) Notation: XY I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
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Correlation Coefficient
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Property Corr.2
• For any constants, a1, b1, a2, and b2,
(1) if a1a2 > 0, then Corr(a1X+b1,a2Y+b2) = Corr(X,Y), (2) if a1a2 < 0, then Corr(a1X+b1,a2Y+b2) = ‐ Corr(X,Y).
Property Corr.1
‐1  Corr(X,Y)  1
• Notes
•
1) If Corr(X,Y) = 0, equivalently Cov(X,Y), then there is no linear relationship between X and Y, and X and Y are uncorrelated random variables.
Note that correlation coefficient does not depend on any units of measurement such as dollars, cents, years quarters, months, and so on.
2) If Corr(X,Y) = 1, there is a perfect positive linear relationship.
3) If Corr(X,Y) = ‐1, there is a perfect negative linear relationship.
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IV. Features of Joint and Conditional Distributions
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IV. Features of Joint and Conditional Distributions
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Variance of Sums of Random Variables Summary: Covariance and Correlation
Covariance
Cov(X,Y) = E(X‐uX)(Y‐uY)
Cov.1 If X, Y independent,
Cov(X,Y) = 0
Correlation
Corr(X,Y) = .
Property VAR.3
• For constants a and b,
,
· .
Var(aX+bY) = a2Var(X) + b2Var(Y) + 2abCov(X,Y)
Corr.1 ‐1  Corr(X,Y)  1
Var(aX‐bY) = a2Var(X) + b2Var(Y) ‐ 2abCov(X,Y)
Corr.2 If a1a2>0, Corr(a1X+b1, a2Y+b2) = Corr(X,Y)
Cov.2 Cov(a1X+b1,a2Y+b2) = a1a2Cov(X,Y)
If a1a2<0, Corr(a1X+b1, a2Y+b2) = – Corr(X,Y)
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
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Variance of Sums of Random Variables A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
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Variance of Sums of Random Variables
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Example:
X denote Friday profits earned by a restaurant Y denote Saturday profits earned by a restaurant Let E(X) = E(Y) = 300 and sd(X) = sd(Y) = 15
If X and Y are uncorrelated, Cov(X,Y)=0, then
1) The variance of sum is the sum of the variance.
Var(X + Y) = Var(X) + Var(Y)
What is the expected profits for the two nights?
– E(Z) = E(X) + E(Y) = 300+300 = 600
2) The variance of difference is the sum of the variance, not the difference in variances.
Var(X ‐ Y) = Var(X) + Var(Y)
If X and Y are independent, what is the standard deviation of total profits?
– Var(Z) = Var(X) + Var(Y) = 2*225 = 450
– Standard deviation = 21.21
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IV. Features of Joint and Conditional Distributions
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IV. Features of Joint and Conditional Distributions
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Variance of Sums of Random Variables A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Variance of Sums of Random Variables
In summation notation, Property VAR.4
•
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
If {X1, …, Xn} are pairwise uncorrelated random variables and {ai : i= 1, …, n} are constants, then
Var(a1X1 + …. + anXn) = a12Var(X1) + … + an2Var(Xn)
•
A special case, if ai=1 for all i, The random variables {X1, …, Xn} are pairwise uncorrelated if each variable in the set is uncorrelated with every other variable in the set.
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
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Variance of Sums of Random Variables A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Let X Binomial(n,).
where Yi are independent Bernoulli() random variables.
Var(X) A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Example: Airline reservations n=120 and =.85
Yi=1, if person i shows up at the airport
X = Y1 + … + Yn
•
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Variance of Sums of Random Variables Example:
•
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
•
= Var(Y1) + … + Var(Yn)
= n(1‐)
What is the variance of the number of passengers arriving for their reservation?
Var(X) = n(1‐) = (120)(.85)(.15) = 15.3
standard deviation = 3.9
• Notes
1) Var(Yi) = (1‐ )] (see IIIB.1)
2) Independent random variables are uncorrelated (see COV.1)
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IV. Features of Joint and Conditional Distributions
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IV. Features of Joint and Conditional Distributions
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Conditional Expectation
Variance and Expectation
Variance: Var(X) = E(X‐x)2
Var(X) = E(X)2 ‐ x2
Expected Value: E(X)
Property VAR.4
•
1, …, Xn} are pairwise uncorrelated random variables and {ai : i= 1, …, n} are E.1 If {X
E(c) = c
v.1
Var(c) = 0
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
•
In economics and business, we would like to explain one variable, called Y, in terms of another variable, say X. •
We have learn the concept of conditional probability function of Y given X. •
Note that the distribution of Y depends on X=x.
constants, then
E.2 E(aX+b) = aE(X) + b
v.2
E.3 E ∑
∑
v.3 If ai=1, E ∑
∑
Var(aX+b) = a2Var(X)
Var(a1X1 + …. + anXn) = a12Var(X1) + … + an2Var(Xn)
•
Var(aX+bY) = a2Var(X) + b2Var(Y) + 2abCov(X,Y)
The random variables {X1, …, Xn} are pairwise uncorrelated if each variable in the set is uncorrelated with every other variable in the set.
Var(aX–bY) = a2Var(X) + b2Var(Y) – 2abCov(X,Y)
v.4 Let {X1, …, Xn} be pairwise uncorrelated. ∑ 2
Var ∑
∑
If ai=1, Var ∑
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IV. Features of Joint and Conditional Distributions
Conditional Expectation
•
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A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Conditional Expectation
•
We can summarize a relationship between Y and X by the conditional expectation of Y given X, sometimes called the conditional mean. 106
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IV. Features of Joint and Conditional Distributions
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
When Y is a discrete random variable taking on values {y1, … , ym}, then
m
E(Y | X  x)   y j fY | X ( y j | x)
j 1
•
We denote the conditional mean by E(YX=x) or E(Yx) for short.
•
E(Yx) is a function x, which tells us how the expected value of Y varies with x.
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
m
E(Y | x)   y j f ( y j | x)
j 1
107
•
When Y is continuous, E(Yx) is defined by integrating yfYX(yx) over all possible values of y.
•
The conditional expectation is simply the weighted average of possible values of Y, but now the weights take into account a particular value of X. I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
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Conditional Expectation
•
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Conditional Expectation
Conditional expectations can be linear and nonlinear functions. •
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Conditional expectations can be linear and nonlinear functions. Example: X is a random variable greater than zero
Example:
– E(Yx) = 10/x
– see graph!
•
E(wageeduc) = 4 +.60educ
•
What is the average wage rate given employees have 16 years of education?
E(wageeduc=16) = 4 +.60(16) = 4 + 9.6 =13.6
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
Notes:
1) This could represent a demand function (Y=quantity; X=price)
2) An analysis of linear association, such as correlation analysis, would be incomplete.
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Properties of Conditional Expectation
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Properties of Conditional Expectation
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Property CE.1
• E[c(X)X] = c(X) for any function c(X).
Property CE.2.
For functions a(X) and b(X), E[a(X)Y+b(X)X) = a(X)E(YX) + b(X)
•
This implies that functions of X behave as constant when we compute the expectation conditional on X.
Example 1: Find the conditional expectation of a function, a+bx
•
Example 1: a+bx
E(a + bxx) = a + bx
Example 2: Find the conditional expectation of a function, a+bx = 4 +.60educ
E(X2X) = X2
•
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Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
4 +.60educ E(4 + .60educeduc) = 4 + .06educ
When we know X, we know X2
Example 3: Find the conditional expectation of a function, wage – 4 – .60educ
wage – 4 – .60educ
E(wage – 4 – .60educ educ) = E(wageeduc) – 4 – .60educ
Example 2: x = educ
E(xx) = x
E(educeduc) = educ
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Example 4: Find the conditional expectation of a function, XY + 2X2
XY + 2X2
E(XY + 2X2 X) = XE(YX) + 2X2
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IV. Features of Joint and Conditional Distributions
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Properties of Conditional Expectation
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Properties of Conditional Expectation
•
Property CE.3:
• If X and Y are independent, E(YX) = E(Y)
Notes:
1) If X and Y are independent, then the expected value of Y given X does not depend
on X.
E[E(YX)] = E(Y) [B.2] 3) A special case is that if U and X are independent and E(U)=0, then E(U) =0, and
E(UX) = 0
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A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
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Properties of Conditional Expectation
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Property CE.5
• If E(Y X) = E(Y), then Cov(X,Y)=0 and Corr(X,Y)=0, or X and Y are uncorrelated.
1. Given E(educ)=11.5 and
Property CE.2
Notes:
1. If knowledge of X does not change the expected value of Y, E(Y), then X
and Y are uncorrelated.
2. If X and Y are correlated, then E(YX) must depend on X.
2. The law of iterated expectation implies that E[E(wageeduc)] = E(wage) = E(4+.60educ) Property CE.4
= 4+.60E(educ) = 4 + (.6)(11.5) = 10.9 or $10.9 an hour
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
Example:
wage = 4 + .60educ
E(wageeduc) = 4 + .60educ [See Problem B.10]
1) Find E(YX) as a function of X. Property CE.2
2) Take the expected value of E(YX) with respect to the distribution of X. 3) End up with E(Y)
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
Properties of Conditional Expectation
The law of iterated expectations says that the expected value of (X) is simply equal to the expected value of Y.
Property CE.4
2) Example: If wage and education are independent, then the average wages of high school and college graduates are the same.
IV. Features of Joint and Conditional Distributions
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
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•
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The converse is not true: If X and Y are uncorrelated, E(YX) could still depend on X. I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
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Expected Value and Conditional Expectation (CE)
Expected Value: E(X)
Property VAR.4
E.1 E(c) = c
•
Conditional Variance
Conditional Expectation
CE.1
•
Given random variables X and Y,
Var(YX=x) = E{[Y‐E(Yx)]2x}
•
The variance of Y, conditional on X=x is the variance associated with the conditional distribution of Y, given X=x.
Var(YX=x) = E{[Y‐E(Yx)]2x}
= E(Y2x) – [E(Yx)]2
E[c(X) X] = c(X) If {X1, …, Xn} are pairwise uncorrelated random variables and {ai : i= 1, …, n} are constants, then
CE.2
E[a(X)Y+b(X)X]
E.2 E(aX+b) = aE(X) + b
= a(X)E[YX]+b(X)
Var(a1X1 + …. + anXn) = a12Var(X1) + … + an2Var(Xn)
E.3 E ∑
∑
CE.3
If X and Y are independent, E(YX)= E(Y)
•If aiThe random variables {X
∑
=1, E ∑
1, …, Xn} are pairwise uncorrelated if each variable in the set is uncorrelated with every other variable in the set.
CE.4
E[E(YX)] = E(Y)
CE.5 If E(YX)= E(Y), Cov(X,Y)=0
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IV. Features of Joint and Conditional Distributions
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Conditional Variance
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Conditional Variance
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Example:
Var(SAVINGINCOME) = 400 + .25INCOME
Property CV.1
• If X and Y are independent, then Var(YX)=Var(Y)
Notes:
1) As income increases, the variance of saving levels also increases.
2) The relationship between variance of SAVING and INCOME is separate from that between the expected value of SAVING and INCOME.
•
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
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This is obvious. If the distribution of Y given X does not depend on X. Var(YX) is just one feature of this distribution.
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IV. Features of Joint and Conditional Distributions
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Problem B.10
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
Problem B.10 continue…
A. Covariance
B. Correlation Coefficient
C. Conditional Expectation
D. Conditional Variance
B.10 Suppose that at a large university, college grade point average, GPA, and SAT score, SAT, are related by the conditional expectation E(GPASAT) = .70 + .002SAT.
(ii) If the average SAT in the university is 1,100, what is the average GPA? [ans.]
(Hint: Use Property CE.4. E[E(YX)] = E(Y)]
(i) Find the expected GPA when SAT 800. Find E(GPASAT = 1,400). Comment on the difference. [ans.]
(iii) If a student’s SAT score is 1,100, does this mean he or she will have the GPA found in part (ii)? Explain. [ans.]
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IV. Features of Joint and Conditional Distributions
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IV. Features of Joint and Conditional Distributions
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Problem B.10 (i)
Problem B.10 (ii)
(i) (ii) • E(GPA|SAT = 800) = .70 + .002(800) = 2.3.
• Following the hint, we use the law of iterated expectations. • Since E(GPA|SAT) = .70 + .002 SAT, Similarly, • E(GPA|SAT = 1,400) = .70 + .002(1400) = 3.5. • The difference in expected GPAs is substantial, and the difference in SAT scores is also rather large.
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IV. Features of Joint and Conditional Distributions
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
• the (unconditional) expected value of GPA is E(GPA) = 70 + .002 E(SAT)
= .70 + .002(1100) = 2.9.
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IV. Features of Joint and Conditional Distributions
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A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
V. The Normal and Related Distributions Problem B.10 (iii)
(iii)
• No, 2.9 is an average GPA. It does not mean a particular person will obtain GPA of 2.9.
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
IV. Features of Joint and Conditional Distributions
Suppose X is a normal random variable. The pdf of X can be written as
•
•
where =E(X) and 2=Var(X). We say that X has a normal distribution with expected value  and variance 2, written as X  Normal(,2)
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V. The Normal and Related Distributions
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
•
The normal distribution and its related distribution is most widely used distribution in statistics and econometrics. V. The Normal and Related Distributions
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
•
We will rely on normal and related distributions to make inference in statistics and econometrics.
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
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The Normal Distribution
•
Notes on the Normal Distribution
127
1)
A normal random variable is a continuous random variable.
3)
2)
Its pdf has a familiar bell shape.
4)
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
A normal distribution is symmetric; thus  is also the median of X.
The normal distribution is sometimes called the Gaussian distribution
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V. The Normal and Related Distributions
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The Normal Distribution
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
The Standard Normal Distribution
 and  read “phi”
•
Certain random variables appear to follow a normal distribution Examples: heights and weights.
•
Some distribution is skewed towards the upper tail, i.e., income. However, a variable can be transformed to achieve normality. A popular transformation is the natural log. Let Y be log of income X.
Y=log(X) We say that X has a lognormal distribution. I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
V. The Normal and Related Distributions
•
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•
Suppose that a random variable Z is normally distributed with zero expected value (=0) and unit variance (2=1). Then, Z has a standard normal distribution. Z  Normal(0,1)
•
The pdf of a standard normal variable, denoted by (z),
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
V. The Normal and Related Distributions
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
The Standard Normal Distribution
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
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The Standard Normal Distribution
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
The standard normal cumulative distribution function, denoted by (z), is obtained by the area under  to the left of z. (see graph)
(z) = P(Z<z)
P(Z<3.1) = .999
– Most software packages include simple commands for computing values of the standard normal cdf.
Example:
– P(Z<‐3.1) = .001
– P(Z<3.1) = .999
P(Z<-3.1) = .001
Note that P(Z>3.1) = .001
- 3.10 │
│ 3.10
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V. The Normal and Related Distributions
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V. The Normal and Related Distributions
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The Standard Normal Distribution
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
•
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
Given (z) = P(Z<z), three important formulas for standard normal cdfs:
1) P(Z>z) = 1‐ (z)
2) P(Z<‐z) = P(Z>z) 3) P(a<Z<b) = (b) – (a)
4) Another useful expression is that, for any c>0,
Notes:
– 1) The probability P(Z>c) is simply twice the probability P(Z>c).
– 2) This reflects the symmetry of the standard normal distribution.
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
V. The Normal and Related Distributions
The Standard Normal Distribution
• Examples:
1) P(Z>.44) 2) P(Z<‐.92) 3) P(‐1<Z<.5) 4) P(│Z│>.44) I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
V. The Normal and Related Distributions
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A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
= 1 ‐ P(Z<.44) = 1 ‐ .67 = .33
= P(Z>.92) = 1‐ P(Z<.92) = 1 ‐ .821 = .179
= P(Z<.5) ‐ P(Z<‐1) = .692 ‐ .159 = .5328
= P(Z>.44) + P(Z<.‐44) + = 2*P(Z>.‐44) = 2*(.33) = .66
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V. The Normal and Related Distributions
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V. The Normal and Related Distributions
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136
Properties of the Normal Distribution A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
Property Normal.1
• If X  Normal(,2), then (X‐)/  Normal(0,1).
• Z  Normal(0,1).
Example:
• Suppose X  Normal(3,4). Compute P(X1)
 x -3

P ( X  1)  P ( X - 3  1 - 3)  P 
 1
 2

= P(Z‐1) = (‐1) = .159
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V. The Normal and Related Distributions
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Probabilities for a Normal Random Variable
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V. The Normal and Related Distributions
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A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
Example:
Compute P(2X 6) when X  Normal(4,9)
 2-4 x-4 6-4 
P(2<X  6)  P 
<


3
3 
 3
2
2
...................  P(  Z  )
3
3
= (.67)‐ (‐.67) = .749 ‐ .251 = .498
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V. The Normal and Related Distributions
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I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
V. The Normal and Related Distributions
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Probabilities for a Normal Random Variable
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
Example:
Compute P(X>2) when X  Normal(4,9
P(X>2) = P(X>2)+P(X<‐2)
X−4 2−4
X−4 2−4
= P[
> ] + P [
< ‐
] 2
= P(Z> ‐ )+P(Z<‐2)
3
2
= 1 ‐ (‐ ) + (‐2)
3
= 1 ‐.251+.023
= .772
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V. The Normal and Related Distributions
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Properties of the Normal Distribution A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
Property Normal.3
• Let X and Y are jointly normally distributed. X and Y are independent if and only if Cov(X,Y)=0.
•
Example: Let Y = 2X + 3
If X  Normal(1,9) then Y  Normal(5,36)
Y
= 2X + 3
E(Y)
= 2E(X) + 3 = 2*1 =5
Var(Y)
= 22Var(X) = 4*9 = 36
sd(Y) = 6
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Properties of the Normal Distribution Property Normal.2
• If X  Normal(,2), then
aX+b  Normal(a+b, a22)
V. The Normal and Related Distributions
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
V. The Normal and Related Distributions
143
Zero correlation and independence are not the same. But in the case of normally distributed random variables, it turns out that zero correlation suffices for independence.
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V. The Normal and Related Distributions
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Properties of the Normal Distribution A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
Properties of the Normal Distribution Property Normal.4
• Any linear combination of independently identically distributed (i.i.d.) normal variables has a normal distribution.
Example:
• Suppose Y1, Y2, … , Yn are i.i.d. normal random variables and each is distributed as Normal(,2). Then
Example:
• Let Xi be i.i.d. normal variables. Note that it is distributed as Normal(,2) Y
= X1 + 2X2 – 3X3
Mean:
E(Y) =  + 2 ‐ 3 = 0
Variance Var(Y) = 2 +42 + 92 = 142
Show:
= (1/n)[Y1+ ….+Yn]
E( ) Var( )
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The Chi‐Square Distribution
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
V. The Normal and Related Distributions
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
= (1/n)[ + … +] = 
= (1/n2)[Var(Y1) + … + Var(Yn)]
= (1/n2)[2 + … + 2] = 2/n
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Notes: The Chi‐Square Distribution
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
•
The chi‐square distribution is obtained from independent, standard normal random variables.
1) A chi‐square random variable is always nonnegative
3) If X  2n then, • the expected value of X is n
• the variance of X is 2n
•
Let Zi, i =1, …, n be independent random variables, each distributed standard normal. Define a new variable X as the sum of the squared of the Zi:
2) It is not symmetric about any point. It is skewed to the right for small df.
4) For df in excess of 100, the variable can be treated as a standard normal variable, where k is the df
•
X has what is known as a chi‐square distribution with n degrees of freedom. – df
•
(the number of terms in the sum)
We write this as X  2n
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
V. The Normal and Related Distributions
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V. The Normal and Related Distributions
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A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
The t Distribution I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
V. The Normal and Related Distributions
Notes: The t Distribution
3) Expected value and Variance:
Expected value = 0 (for n>1)
Variance = n/(n‐2) (for n>2)
2) t distribution gets its degrees of freedom from the chi‐square variable.
4) t distribution is symmetric about its mean, but has more area in tails.
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
•
The t distribution is the workhorse in classical statistics and multiple regression analysis.
•
Let Z have a standard normal distribution and X has a chi‐square distribution with n degrees of freedom. Assume that Z and X are independent. Then, the random variable T is
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
V. The Normal and Related Distributions
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
1) The random variable T has t distribution with n degrees of freedom.
– We denote this as T  tn
V. The Normal and Related Distributions
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A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
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A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
151
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V. The Normal and Related Distributions
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A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
The F Distribution •
A F distribution is used for testing hypothesis in the context of multiple regression analysis.
•
Let X1k1 and X2k2 and assume that X1 and X2 are independent. Then, the random variable F
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
has a F distribution with (k1,k2) DFs.
• We denote this as FFk1,k2
k1 – the numerator degrees of freedom because it is associated with chi‐
square variable in the numerator.
k2 – is the denominator degrees of freedom
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
Notes: The F Distribution 1) Relationship between t and F statistic
Like the chi‐square distribution, the F distribution is skewed to the right.
2) Relationship between 2and F statistic
1
as k2  
4)
As k1 and k2 become large, the F distribution approaches the normal distribution.
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
V. The Normal and Related Distributions
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
V. The Normal and Related Distributions
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
3)
t2k2 = F1,K2
Fk1,K2 = 153
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
V. The Normal and Related Distributions
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A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
155
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V. The Normal and Related Distributions
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Notes: The F Distribution A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
Problem B.2
In passing, note that since for large df (or n  ),
the t, chi-square, and F distributions approaches the normal distribution.
A. Normal
B. Standard Normal
C. Chi Square
D. t distribution
F. F distribution
B2. Let X be a random variable distributed as Normal(5,4). Find the probabilities of the following events:
(i) P(X  6) [ans.]
(ii) P(X > 4) [ans.]
(iii) P(X – 5> 1) [ans.]
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V. The Normal and Related Distributions
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V. The Normal and Related Distributions
Problem B.2 (i)
Problem B.2 (ii)
(i) Normal (5,4)
(ii) Normal (5,4)
• P(X  6) = P[(X – 5)/2  (6 – 5)/2]
= P(Z  .5) (See Table G.1)
 .692 • P(X > 4) = P[(X – 5)/2 > (4 – 5)/2]
= P(Z > .5)
= 1 ‐ P(Z  .5) (See Table G.1)
= 1 – 0.3085
 .692. where Z denotes a Normal (0,1) random variable. [We obtain P(Z  .5) from Table G.1.] I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
V. The Normal and Related Distributions
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V. The Normal and Related Distributions
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Problem B.2 (iii)
Table G.1
B.1(i)
B.2(ii)
B.3(iii)
(iii) Normal (5,4) • P(|X – 5| > 1)
= P(X – 5 > 1) + P(X – 5 < –1)
= P(X > 6) + P(X < 4)
= P[(X‐5)/2 > (6‐4)/2] + P[(X‐5)/2 < (4‐6)/2)
= P(Z > .5) + P(Z < ‐.5)
•
P(|X – 5| > 1)
•
P(|X – 5| > 1)
= P(Z < ‐.5) + P(Z < ‐.5)
= .3085 +.3085 = .617 (Table G.1)
= P(Z > .5) + P(Z < ‐.5)
= (1 – .692) + (1 – .692) = .616
where we have used answers from parts (i) and (ii). I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
V. The Normal and Related Distributions
Fundamentals of Probability . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat
I. Random Variables II. Joint & Conditional D’s III. Probability D’s IV. Features of Joint & Conditional D’s V. Normal & Related D’s
V. The Normal and Related Distributions
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