Download Document

Keep Life Simple!
We live and work and dream,
Each has his little scheme,
Sometimes we laugh; sometimes we cry,
And thus the days go by.
1
Random Variables
Types
of random variables
Expected values
Binomial and Normal
distributions
2
What Is a Random Variable?
The numerical outcome of a random
circumstance is called a random variable.
Eg. Toss a dice: {1,2,3,4,5,6}
Height of a student
 A random variable (r.v.) assigns a number to
each outcome of a random circumstance.
Eg. Flip two coins: the # of heads

3
Types of Random Variables
A continuous random variable can take any
value in one or more intervals.
** give examples

A discrete random variable can take one of
a countable list of distinct values.
** give examples

4
Distribution of a Discrete R.V.



5
X = a discrete r.v.
k = a number X can take
The probability distribution function (pdf) of X
is:
P(X=k)
How to Find the Function pdf
1.
2.
3.
4.
5.
6
List all outcomes (simple events) in S
Find the probability for each outcome
Identify the value of X for each outcome
Find all outcomes for which X=k, for each
possible k
P(X=k) = the sum of the probabilities for all
outcomes for which X=k
Example: Flip a Coin 3 Times
** find pdf
** draw a plot of pdf
7
CDF of a R.V.

8
The cumulative distribution function (cdf) of X
is:
P(X<k)= sum of P(X=h) over h<k
Example: Flip a Coin 3 Times
** find cdf
** draw a plot of cdf
9
Important Features of a Distribution



10
Overall pattern
Center – mean
Spread – variance or standard deviation
Expected Value (Mean)

11
The expected value of X is the mean
(average) value from an infinite # of
observations of X
Finding Expected Value




X = a discrete r.v.
{ x1, x2, …} = all possible X values
pi is the probability X = xi where i = 1, 2, …
The expected value of X is:
  E ( X )   xi pi
i
12
Example: Flip a Coin 3 Times
** find the mean value
13
Variance & Standard Deviation


Notations as before
Variance of X:
V ( X )     ( xi   ) pi
2
2
i

Standard deviation (sd) of X:

2
(
x


)
pi
 i
i
14
Example: Flip a Coin 3 Times
** find the variance and sd
15
Binomial Random Variables


16
Binomial experiments:
Repeat the same trial of two possible
outcomes (success or failure) n times
independently
The # of successes out of the n trials is
called a binomial random variable
Examples:
Flip a fair coin 3 times (or flip 3 fair coins)
 The # of defective memory chips of 50 chips
 An experimental
treatment for bird flu
 Others?

17
PDF of a Binomial R.V.
p = the probability of success in a trial
 n = the # of trials repeated independently
 X = the # of successes in the n trials
For k = 0, 1, 2, …,n,

P(X=k) =
18
n!
k
nk
p (1  p)
k!(n  k )!
Example: Pass or Fail
Suppose that for some reason, you are not
prepared at all for the today’s quiz. (The quiz is
made of 5 multiple-choice questions; each
has 4 choices and counts 20 points.)
You are therefore forced to answer these
questions by guessing. What is the probability
that you will pass the quiz (at least 60)?
19
Mean & Variance of a Binomial R.V.

Notations as before

Mean is

20
  E ( X )  np
Variance is
  V ( X )  np(1  p)
2
Distribution of a Continuous R.V.

The probability density function (pdf) for a
continuous r.v. X is a curve such that
P(a < X <b) =
the area under it over the interval [a,b].
21
Normal Distribution



22
The “model” distribution of a continuous r.v.
The r.v. with a normal distribution is called a
normal r.v.
The pdf of a normal r.v. looks like:
CDF of a Normal R.V.

X: a normal r.v. with mean  and
standard deviation 
F(a) = P(X < a)
= P( X    a   )


= see Table A.1
23
z score
Example



24
Suppose that the final scores of ST1000
students follow a normal distribution with  =
70 and  = 10. What is the probability that a
ST1000 student has final score 85 or above
(grade A)?
Between 75 and 85 (grade B)?
Below 50 (F)?