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1/09 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 PROBABILITY AND STATISTICS II Assignment 1 Section A 1. An experiment with a binary outcome (success or failure) is repeatedly and independently conducted. Let p (0 , 1) be the probability of success. Define a random variable X to be the number of trials required to obtain the r th success. (a) Explain why the random variable X can be written as r i =1 X i , where X 1 , . . . , X r are i.i.d. random variables, each having the mass function f ( x ) = (1 - p ) x - 1 p, x = 1 , 2 , . . . . (b) Find the moment generating function of X 1 as defined in (a). (c) Deduce from (a) and (b) the moment generating function of X . 2. Let X i be a Poisson random variable with mean λ i > 0, for i = 1 , . . . , n . Assume that X 1 , . . . , X n are independent. (a) Find the moment generating function of X i . (b) Using (a) or otherwise, derive the moment generating function of S = n i =1 X i . What is the sampling distribution of S ? 3. Let X 1 , . . . , X n be independent normal random variables such that X i has mean μ i and variance σ 2 i . It is known that an N ( μ, σ 2 ) random variable has the moment generating function M ( t ) = exp( μt + σ 2 t 2 / 2).
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