{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture7

# lecture7 - ISYE 2028 A and B Lecture 7 Conditional...

This preview shows pages 1–4. Sign up to view the full content.

ISYE 2028 A and B Lecture 7 Conditional Expectation and Prediction Dr. Kobi Abayomi February 10, 2009 1 Conditional Expectation (again) We should recall the definition of the conditional expectation: E ( X | Y = y ) = x x P ( x | Y = y ) , discrete R xf X | Y = y dx, continuous (1) In the case that (either) p y , f y are functions of only y (i.e. when Y 6 = g ( X ): E ( X | Y = y ) = ( P x x P ( x,y ) p y , discrete R xf X | Y = y dx f Y , continuous (2) If you are in doubt — i.e. you are trying to solve a problem — calculate f Y = R f X,Y dx explicitly. 1.1 Example Let X, Y f X,Y = e - x/y e - y y · 1 { x,y (0 , ) } . Find the conditional expectation E ( X | Y ). Well: 1

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

View Full Document
f Y = Z 0 e - x/y e - y y dx = e - y Z 0 1 y e - x/y dx = e - y Which yields... f X | Y = e - x/y e - y y · 1 e - y = 1 y e - x/y So, the conditional expectation is: E ( X | Y = y ) = Z 0 x y e - x/y dx But this is just the expectation of an exponential random variable with parameter λ = 1 y . So E ( X | Y ) = y . 2 Computing Expectations by Conditioning Recall: E ( X ) = E ( E ( X | Y )); the expectation of the conditional expectation is the original, or unconditioned expectation. We saw a proof of this in an earlier lecture. This fact allows us to compute expectations by conditioning on any random variable (which makes the computation easy). 2.1 Example: Expectation of a Random Sum Let X 1 , ..., X N with N a random number and X i N . Find the expectation of the random sum E ( N i =1 X i ). Condition on N : 2
E ( N X i =1 X i ) = E [ E ( N X i =1 X i | N = n )] = E [ N · E ( X )] = E ( N ) · E ( X ) 2.2 Example: ρ in N 2 is correlation Let X, Y N 2 ( μ X , μ Y , σ 2 X , σ 2 Y , ρ ), the bivariate normal distribution. We’ve defined the correlation ρ X,Y = E ( XY ) - μ X μ Y σ X σ Y

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

View Full Document
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