Download Lecture 23

AMS 311
April 25, 2000
Chapter Eight
Bivariate Distributions
Homework: Due May 2:
Chapter 8: p. 301: 2, 10; p 313: 2; p 326: 6, 10; p 360: 4, 6.
Chapter 9: p. 380: 3, 4; p 389: 1, 3 .
Bivariate Distributions:
Joint probability function (discrete case) and joint probability density function.
Example problem:
Let the joint probability density functions of the random variables X and Y be given by
f ( x, y)  2 if 0  y  x  1, and zero elsewhere.
Calculate the marginal density function of X and Y.
Calculate P( X  2Y ).
Joint probability function and joint probability density function of two independent
random variables.
If X and Y are independent random variables, then
E ( g ( X )h(Y ))  E ( g ( X )) E (h(Y )).
Conditional Distributions
Recall the definition of conditional probability. Conditional probability distributions of
discrete random variables are defined in a natural extension:
p( x , y )
p X |Y ( x| y) 
.
p( x )
The definition of a conditional probability density function is made by analogy:
f ( x, y)
f X |Y ( x| y) 
.
f ( x)
Example problem:
For the pdf in the first example problem, find f X |Y ( x| y).
The conditional distribution is a probability distribution. As such, it has moments such as
the mean and the variance. The conditional mean is the mean of the conditional
distribution, and the conditional variance is the variance of the conditional distribution.
Transformation of two random variables is a crucial problem and hard to handle. It is
important to review your multivariable calculus so that you are up to speed technically.
The probability theory is not hard. Consider the following example problem:
Let X and Y be independent random variables with common probability density function
f ( x)  e  x , x  0, and zero otherwise. Find the joint probability density function of
U=X+Y and V=eX.