{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 09_17 - STAT 410 Examples for Fall 2008 2.3 1 Conditional...

This preview shows pages 1–3. Sign up to view the full content.

STAT 410 Examples for 09/17/2008 Fall 2008 2.3 Conditional Distributions and Expectations. 1. Consider the following joint probability distribution p ( x , y ) of two random variables X and Y: y x 0 1 2 p X ( x ) 1 0.15 0.15 0 0.30 2 0.15 0.35 0.20 0.70 p Y ( y ) 0.30 0.50 0.20 f) Find the conditional probability distributions p X | Y ( x | y ) = ( ) ( ) y p y x p , Y of X given Y = y , conditional expectation E ( X | Y = y ) of X given Y = y , conditional variance Var ( X | Y = y ) of X given Y = y , E ( E ( X | Y ) ) , and Var ( E ( X | Y ) ) . x p X | Y ( x | 0 ) x p X | Y ( x | 1 ) x p X | Y ( x | 2 ) 1 0.15 / 0.30 = 0.50 1 0.15 / 0.50 = 0.30 1 0.00 / 0.20 = 0.00 2 0.15 / 0.30 = 0.50 2 0.35 / 0.50 = 0.70 2 0.20 / 0.20 = 1.00 E ( X | Y = 0 ) = 1.5 E ( X | Y = 1 ) = 1.7 E ( X | Y = 2 ) = 2.0 Var ( X | Y = 0 ) = 0.25 Var ( X | Y = 1 ) = 0.21 Var ( X | Y = 2 ) = 0.00 E ( X | Y = y ) p Y ( y ) Var ( X | Y = y ) p Y ( y ) 1.5 0.30 0.25 0.30 1.7 0.50 0.21 0.50 2.0 0.20 0.00 0.20 E ( E ( X | Y ) ) = 1.7 = E ( X ) E ( Var ( X | Y ) ) = 0.18 Var ( E ( X | Y ) ) = 0.03 < 0.21 = Var ( X ) . 0.21 = 0.03 + 0.18.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Def Var ( X | Y ) ) = E [ ( X – E ( X | Y ) ) 2 | Y ] = E ( X 2 | Y ) [ E ( X | Y ) ] 2 Theorem E ( E ( X | Y ) ) = E ( X ) Var ( E ( X | Y ) ) Var ( X ) Furthermore, Var ( X ) = Var ( E ( X | Y ) ) + E ( Var ( X | Y ) ) g) Find the conditional probability distributions p Y | X ( y | x ) = ( ) ( ) x p y x p , X of Y given X = x , conditional expectation E ( Y | X = x ) of Y given X = x , conditional variance Var ( Y | X = x ) of Y given X = x , E ( E ( Y | X ) ) , and Var ( E
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern