Download Supplementary Note (Ch3)

MAT 446
Supplementary Note for Ch 3
Myung Song, Ph.D.
© 2009 W.H. Freeman and Company
Random Variables
Definition
For a given sample space S of some experiment, a
random variable (rv) is any rule that associates a number
with each outcome in S .
In mathematical language, a random variable is a function
whose domain is the sample space and whose range is the
set of real numbers.
1.2
Random Variables
Example
1. Flip a fair coin once
S={H, T}
X: # of heads
X(H)=1, X(T)=0
Any random variable whose only possible values are 0 and
1 is called a Bernoulli rv.
2. Free Throws (4 times)
S={HHHH, HHHM, … MMMH, MMMM}
X: # of hits then X= 0, 1, 2, 3, 4
X(MMMM) = 0, X(HHMM) = 2, X(HHHM) = 3, …
Two Types of Random Variables
Discrete rv (in Ch3)
A discrete random variable is an rv whose possible
values either constitute a finite set or else can be listed in
an infinite sequence in which there is a first element, a
second element, and so on.
ex) Examples in the previous page.
1.4
Two Types of Random Variables
Continuous rv (in Ch4)
A random variable is continuous if both of the following apply:
1. Its set of possible values consists either of
(1) all numbers in a single interval on the number line (possibly infinite
in extent) or
(2) all numbers in a disjoint union of such intervals.
ex) (1, 5), [2, 7] (10, 100)
2. No possible value of the variable has positive probability, that is,
P(X = c) = 0 for any possible value c.
Probability Distributions for Discrete rv
Definition
Parameter
Definition
Suppose p(x) depends on a quantity that can be assigned
any one of a number of possible values, with each different
value determining a different probability distribution. Such a
quantity is called a parameter of the distribution. The
collection of all probability distributions for different values
of the parameter is called a family of probability
distributions.
Tip:
- Parameters are quantities explaining the characteristics of
the distributions
- They are associated with central location, variation,
skewness, kurtosis,…..
Cumulative Distribution Function
Definition
The cumulative distribution function (cdf) F(x) of a
discrete rv X with pmf p(x) is defined for every number x by
For any number x, F(x) is the probability that the observed
value of X will be at most x.
Cumulative Distribution Function
Proposition
Cumulative Distribution Function
Properties
1.
lim F ( x)  0 and lim F ( x)  1
x 
x 
2. F(x) is non-decreasing.
3. F(x) is right continuous i.e.
lim F ( x)  F (a)
xa 
3.3 Expected Values of Discrete
Random Variables
The Expected Value of X
Definition
The Expected Value of X
Ex) (after example 3.15)
Let X be the number of credit cards owned by a randomly
selected adults. The corresponding pmf is following:
x
0
1
2
3
P(x)
0.1
0.2
0.4
0.3
What is the expected number of credit cards?
The Expected Value of X
Ex) Geometric Distribution (after Example 3.18)
0.6 x 10.4
p( x)  
0
x  1, 2,...
o.w.
What is the expected value of X?
The Expected Value of a function
Proposition
The Expected Value of a function
Proposition
The Expected Value of a function
Ex) Credit Cards example – Continued (after example 3.22)
(1)
What is the expected value of X-1?
(2)
What is the expected value of 2X?
(3)
What is the expected value of 2X+3?
The Variance of X
Definition
Let X have pmf p(x) and expected value m. Then the
2
2
variance of X, denoted by V(X) or  X , or just 
The standard deviation (SD) of X is
The Shortcut Formula for Variance
Proposition
The Variance of X
Ex) Credit Cards
Let X be the number of credit cards owned by a randomly
selected adults. The corresponding pmf is following:
x
0
1
2
3
P(x)
0.1
0.2
0.4
0.3
What is the variance of credit cards?
(Hint: E(X) = 1.9)
The Rules of Variance
Proposition
The Variance of X
Ex) Credit Cards example – Continued
(1)
What is the variance of X-1?
(2)
What is the variance and the standard deviation of 2X?
(3)
What is the variance and the standard deviation of
2X+3?
3.4 Moments and Moment Generating
Functions
Moment
Definition
The k moment of X about   E[( X -  ) ]
th
k
The k moment of X (about 0)  E[ X ]
th
Examples
Why moments?
k
Moment Generating Function
Definition
The moment generating function (mgf) of a discrete
random variable X is defined to be
where D is the set of possible X values.
(note: Mx(t) is a function of t NOT x )
We will say that the mgf exists if it is defined for an open
interval including 0.
Moment Generating Function
Examples
- Example 3.26
- Bernoulli r.v. (related to Example 3.27)
1  p

p( x)   p
0

if x  0
if x  1
o.w.
- Geometric Dist’n (related to Example 3.29)
(1  p ) x 1 p
p( x)  
0
x  0,1, 2,...
o.w.
Moment Generating Function
Proposition
If the mgf exists and is the same for two distributions, then
the two distributions are the same.
That is, the mgf uniquely specifies the probability
distribution; there is a one-to-one correspondence between
distributions and mgf’s.
Example
Moment Generating Function
Theorem
For example,
Moment Generating Function
Example 3.31
Ex)
Let X be the number of defectives by a randomly selected
machine. The corresponding pmf is following:
X
0
1
2
P(x)
0.6
0.3
0.1
Calculate E(X) and V(X) by using mgf.
Moment Generating Function
Ex) Geometric Distribution (related to Example 3.32)
(1  p) x 1 p
p( x)  
0
x  1, 2,3,...
o.w.
What is the mgf of X?
Calculate E(X) and V(X) by using mgf.
Cummulant Generating Function
Definition
The Cummulant Generating Function of X, Rx(t) is
defined as:
Rx (t )  ln[ M x (t )]
Properties
Cummulant Generating Function
Ex) Geometric Distribution (related to Example 3.33)
(1  p) x 1 p
p( x)  
0
x  1, 2,3,...
o.w.
What is the cummulant generating function of X?
Calculate E(X) and V(X) by using cummulant generating
function
Binomial Probability Distribution
THE BINOMIAL SETTING
1. The number of trials n is fixed before
experiments
2. For each trial, only two results “success” and
“failure.”
3. The trials are independent.
4. The probability of a success p is fixed.
Binomial rv and Distribution
Definition
Given a binomial experiment consisting of n trials, the
binomial random variable X associated with this
experiment is defined as
X = the number of successes among the n trials
Notation
Because the pmf of a binomial rv X depends on the two
parameters n and p, we denote the pmf by b(x; n, p).
We will often write X ~ Bin(n, p) to indicate that X is a
binomial rv based on n trials with success probability p.
Binomial rv and Distribution
 Our first step in finding a formula for the binomial pmf that
a binomial rv takes any value is adding probabilities for
the different ways of getting exactly that many successes
in n trials.
Binomial Coefficient
 The number of ways of arranging k successes among n
trials is given by the binomial coefficient,
𝑛
𝑛!
=
, read “n choose k” (if nothing else, to
𝑘! 𝑛−𝑘 !
𝑘
𝑛
distinguish it from the fraction ), for k = 0, 1, 2, …, n.
𝑘
 Note: factorial notation,
𝑛! = 𝑛 𝑛 − 1 𝑛 − 2 ∙ ⋯ ∙ 3 2 1
0! = 1
Binomial rv and Distribution
 The binomial coefficient counts the number of different
ways in which x successes can be arranged among n
trials. The pmf b(x: n, p) is this count multiplied by the
probability of any one specific arrangement of the x
successes.
Binomial Probability
n x
n x
b( x; n, p )    p (1  p )
 x
Number of
arrangements of x
successes
Probability of x
successes
Probability of n-x
failures
Binomial rv and Distribution
Notation
Expectation and Variance of X
Proposition
If X ~ Bin(n, p), then
E ( X )  np
V ( X )  np (1  p )
 X  np(1  p)
Moment Generating Function of X
Proposition
If X ~ Bin(n, p), then
M X (t )  ( pe  1  p )
t
n
Poisson Probability Distribution
Introduction to Poisson Distribution
The Poisson distribution is popular for modeling the
number of times an (rare) event occurs in an interval of
time or space.
Example
 The number of meteors greater than 1 meter
diameter that strike earth in a year.
 The number of patients arriving in an emergency
room between 11 and 12 pm
 The number of London bombing in a week during
WW2.
Poisson Probability Distribution
Definition
A random variable X is said to have a Poisson
distribution with parameter  ( 0) if the pmf of X is
As
and
such that the mean value
remains constant
Poisson Distribution as a Limit
Proposition
The Mean, Variance and MGF of X
Proposition
If X ~ Poisson( ), then
and