Download Lecture 16 - Stony Brook AMS

AMS 311, Lecture 16
April 3, 2001
Reminder of two problem quiz on April 12 (week from Thursday) and examination 2 on
April 17. The amount of material you are responsible for is growing!
Quiz problems and review
Problem set A.
Let X be a random variable with density function (pdf) f ( x) 
4x 3
, 1  x  2, and zero
15
otherwise.
Find the distribution function (cdf) of X.
Find the density function of the random variable Y  X 2 .
Find Pr{125
.  X  175
. }. This is a review problem from AMS310.
Find E ( X ) and var( X ). This is another review problem.
Problem set B.
Let X be a random variable with density function (pdf) f ( x)  05
. e  0.5x , x  0, and zero
otherwise.
Find E (e 4 X ). This is the quiz problem: 20 points for correct specification of expectation.
The expectation does not exist; that is, the integral does not converge to a finite value.
Not every problem has a well defined answer.
Find E (e Xt ).
Example (problem 4 on page 199):
The probability that a power hand is a full house is 0.0014, What is the probability that in
500 random poker hands there are at least two full houses?
Exponential Random Variables
The density function of an exponential random variable is given by f (t )   e  t , t  0.
1
1
Then, if X is an exponential random variable, E ( X )    , and var( X )   2  ( ) 2 .


The choice of the parameter  is particularly advantageous for Poisson process problems.
Gamma Distribution
The density function of a gamma distribution is given by f ( x)   e
 x
(  x ) n 1
, x  0.
(n  1)!
It is said to have parameters ( n,  ). Characterization of the time until the nth event
occurs in a Poisson process.
The parameter n need not be an integer. The density function can be generalized to
r
r
(  x ) r 1
f ( x )   e  x
, x  0. Then E ( X )  , and var( X )  2 .


 (r )
Chi-squared distributions are a special case of gamma distributions.
Beta Distribution
There is a function called the beta function:
1
 ( ) (  )
B( ,  )   x  1 (1  x)  1 dx 
.
0
 (   )
A random variable X has the beta distribution if its density function is given by
1
f ( x) 
x  1 (1  x)  1 , 0  x  1.
B( ,  )
Chapter Eight: Bivariate Distributions
Definition of joint probability function of two discrete random variables:
p( x, y)  P( X  x, Y  y). One can calculate the marginal probability function of X as
well as the marginal probability function of Y.
There is a parallel definition for continuous random variables. We can define the joint
probability function for the pair of random variables (X, Y).