Example_class_10_handout2

Example_class_10_handout2 - THE UNIVERSITY OF HONG KONG...

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

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Example_class_10_handout2 - THE UNIVERSITY OF HONG KONG...

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