Example_class_10_handout2

# Example_class_10_handout2 - THE UNIVERSITY OF HONG KONG...

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

THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 10 Review Conditional distribution and conditional expectation For any two events E and F, the conditional probability of E given F is deﬁned by P ( E | F ) = P ( E F ) P ( F ) provided that P ( F ) > 0 . Let ( X,Y ) be a discrete bivariate random vector with joint pmf P ( X = x,Y = y ) = p ( x,y ) and marginal pmfs p X ( x ) and p Y ( y ). The conditional pmf of Y given that X = x is the function of y denoted by p Y | X ( y | x ), where p X ( x ) > 0 p Y | X ( y | x ) = P ( Y = y | X = x ) = P ( Y = y,X = x ) P ( X = x ) = p ( x,y ) p X ( x ) . If X is independent of Y , then the conditional pmf becomes p Y | X ( y | x ) = p ( x,y ) p X ( x ) = p X ( x ) p Y ( y ) p X ( x ) = p Y ( y ) For continuous random variables, the conditional distributions are deﬁned as: f Y | X ( y | x ) = f ( x,y ) f X ( x ) provided that f X ( x ) > 0 f X | Y ( x | y ) = f ( x,y ) f Y ( y ) provided that f Y ( y ) > 0 Deﬁnitions and Formulas: (a) Conditional distribution function of Y given X = x : F Y | X ( y | x ) = P ( Y y | X = x ) = i y p Y | X ( i | x ) discrete case ´ y -∞ f Y | X ( t | x ) dt continuous case 1

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

View Full Document
(b) Conditional expectation of g ( Y ) given X = x : E ( g ( Y ) | X = x ) = i g ( i ) p Y | X ( i | x ) discrete case ´ -∞ g ( y ) f Y | X ( y | x ) dy continuous case (c) Conditional mean of Y given X = x : E ( Y | X = x ). (d) Conditional variance of
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 01/16/2012 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

### Page1 / 6

Example_class_10_handout2 - THE UNIVERSITY OF HONG KONG...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online