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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 8 Review Conditional Distributions 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) > Let ( X,Y ) be a discrete bivariate random vector with joint pmf 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 ) > 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 ) > f X  Y ( x  y ) = f ( x,y ) f Y ( y ) provided that f Y ( y ) > Definitions and Formula: (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 R y∞ f Y  X ( t  x ) dt continuous case 1 (b) Conditional Expectation Function of g(Y) given X = x E ( g ( Y )  X = x ) = ∑ i g ( i ) p Y  X ( i  x ) discrete case R + ∞∞ g ( t ) f Y  X ( t  x ) dt continuous case (c) Conditional Mean of Y given X = x : E ( Y  X = x ) (d) Conditional Variance of Y given X = x V ar ( Y  X = x ) = E (( Y E ( Y  X = x )) 2  X = x ) = E ( Y 2  X = x ) ( E ( Y  X = x )) 2 (e) Computing Expectations by Conditioning E ( X ) = E ( E ( X  Y )) V ar ( X ) = E ( V ar ( X  Y )) + V ar ( E...
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This note was uploaded on 05/04/2011 for the course STAT 1301 taught by Professor Smslee during the Spring '08 term at HKU.
 Spring '08
 SMSLee
 Statistics, Conditional Probability, Probability

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