NotesNov1 - X x i E Y | X = x i p X x i if X is discrete with pmf p X EY = E E Y | X = Z ∞-∞ E Y | X = x f X x dx if X is continuous with pdf f

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
Example : Let ( X,Y ) be a continuous random vector with a joint pdf f X,Y ( x,y ) = 15 xy 2 , 0 x,y 1 ,y x. Compute the conditional densities f X | Y and f Y | X ; Compute the conditional mean and the con- ditional variance of Y given X = x for 0 < x < 1.
Background image of page 1

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

View Full DocumentRight Arrow Icon
Computation of expectation and variance via conditioning Using conditional distributions can sometimes simplify computation of means and variances. The idea: the conditional mean E ( Y | X = x ) and the conditional variance Var( Y | X = x ) are functions of the value x of the ran- dom variable X . Since both the conditional mean and the conditional variance are functions of a ran- dom variable X , they can themselves be viewed as random variables.
Background image of page 2
Suppose that X and Y are continuous, with a joint pdf f X,Y . EY = Z -∞ Z -∞ y f X,Y ( x,y ) dx dy = Z -∞ E ( Y | X = x ) f X ( x ) dx.
Background image of page 3

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

View Full DocumentRight Arrow Icon
Abbreviated expression : EY = E ( E ( Y | X )); it is true for any random variables ( X,Y ): dis- crete, continuous or mixed. Terminology : the formula of double expecta- tion .
Background image of page 4
The exact meaning of the formula of double expectation: EY = E ( E ( Y | X )) =
Background image of page 5

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

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: X x i E ( Y | X = x i ) p X ( x i ) if X is discrete with pmf p X ; EY = E ( E ( Y | X )) = Z ∞-∞ E ( Y | X = x ) f X ( x ) dx, if X is continuous with pdf f X . Example : The number of claims arriving to an insurance company in a week is a Pois-son random variable N with mean 20. Assume that the amounts of different claims are inde-pendent exponentially distributed random vari-ables with mean 800. Assume that the claim amounts are indepen-dent of the number of claims arriving in a week. Find the expected total amount of claims the company receives in a week. • The formula of the double expectation EY = E ( E ( Y | X )) is a device to compute expectations by con-ditioning . • There is a device to compute variances by conditioning : for any two random variables X and Y Var Y = E (Var( Y | X )) + Var( E ( Y | X )) ....
View Full Document

This note was uploaded on 12/03/2010 for the course OR 3500 taught by Professor Samorodnitsky during the Fall '09 term at Cornell University (Engineering School).

Page1 / 7

NotesNov1 - X x i E Y | X = x i p X x i if X is discrete with pmf p X EY = E E Y | X = Z ∞-∞ E Y | X = x f X x dx if X is continuous with pdf f

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online