10/31/2012

Chaper 6 Definitions

• Joint distributions
• Marginal distributions
• Conditional distributions
• Conditional means
• Iterated expectation
• Expectation for functions of several variables
• Special Case: when X and Y are independent

Example - Let (X, Y) have the joint distribution defined by the following table:

		X
		1	2	3
Y	1	.2	0	.4
	2	0	.3	.1

• Find P(X+Y>3).
• Find the distribution of X+Y.
• Find the marginal distributions of X and Y.
• Find the conditional distribution of Y given X=3.

Class Exercise - Let (X, Y) have the joint distribution defined by the following table:

		Y
		0	1	2
X	1	.3	.2	.1
	2	.1	.1	.2

• Find the conditional mean of Y given X=1.
• Find the mean of g(X)=X².
• The marginal probability that Y=0 is f_Y(0)=.4.  Show that E[f(0|X)]=.4.
• Show that the conditional means of Y given X=x are: E[Y|X=1]=2/3 and E[Y|X=2]=5/4.
• Use the conditional means above to show that E[E(Y|X)]=E(Y).
• Let g(X,Y)=XY.  Find E(XY).

Please read Section 6.4 for class on Friday.