Assignment 5

Assignment 5 - SMSL/07 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 Assignment 5 Section A 1. Let X,Y be two discrete random variables with their joint mass function f ( x,y ) tabulated below: x \ y 1 2 3 4 1 0.05 0.01 0.10 0.04 2 0.15 0.20 0.05 0.10 3 0.10 0.05 0.10 0.05 (a) Calculate the conditional mass function and cdf of X given Y = 1. (b) Calculate the conditional mass function of X given Y is even. (c) Calculate the conditional expectation μ ( x ) E [ Y | X = x ] for x = 1 , 2 , 3. (d) Calculate the conditional variance ν ( x ) Var( Y | X = x ) for x = 1 , 2 , 3. (e) Calculate E [ μ ( X )] and Var( μ ( X )). (f) Calculate E [ ν ( X )]. (g) Show that Var( Y ) = E [ ν ( X )] + Var( μ ( X )). (h) Calculate the mass function of X - Y . (i) Calculate the conditional mass function of X given X - Y = - 1. (j) Calculate the conditional mean and variance of X given X - Y = - 1. 2. Two cables are connected to a telephone exchange so that calls may come in through either cable. Let X and Y be the numbers of telephone calls passing through the first and second cables respectively on a single day. Assume that X and Y are independent and have Poisson distributions with parameters λ 1 and λ 2 respectively. (a) What is the distribution of
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Assignment 5 - SMSL/07 THE UNIVERSITY OF HONG KONG...

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