5 Assignment 5 (Due Date December 3, 2010)

5 Assignment 5 (Due Date December 3, 2010) - 10/11 THE...

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10/11 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 Probability & Statistics I Assignment 5 Due Date: December 3, 2010 (Hand in your solutions for Questions 1, 5, 18, 23, 27, 34, 37, 38) 1. Let and Y be a continuous random variables distributed in ( with pdf ( 1 , 0 ~ U X ) ) 1 , 0 ( ) y f Y and cdf . Suppose X and Y are independent. Denote as an indicator variable such that if A occurs and otherwise. () y F Y 1 = A I A I 0 = A I (a) Show that, for , 1 0 < < t t y t y I I y t y Y Y t X P < + = = |. (b) Using the result in part (a), show that the cdf of XY W = is given by () () < < + = 1 1 1 0 0 0 1 t t dy y f y t t F t t F t Y Y W . (c) Using variable substitution and integral by parts for the integral in the expression of in part (b), show that t F W ( ) t F W can be also expressed as < < + = 1 1 1 0 0 0 1 t t dx x t F t t t F t Y W . (d) Derive the result in part (c) by first showing that t x t x Y I I x t F x X X t Y P < + = = | f o r 1 0 < < t . (e) Find the cdf of if the cdf of Y is given by XY W = < < = 1 1 1 0 0 0 2 y y y y y F Y . P. 1
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10/11 2. Let , and define Y to be the integer part of () 1 ~ Exp X 1 + X . (a) Find the pdf of Y . What well-known distribution does Y have? (b) Find the conditional distribution of 4 X given . 5 Y 3. Suppose ( ) p n b p X , ~ | , ( ) β α , ~ Beta p . (a) Find the marginal pdf of X . ( X is said to have a beta-binomial distribution.) (b) Find the condition distribution of p , given that x X = . 4. The joint density function of X and Y is given by ( ) > > = + otherwise 0 0 , 0 if , 1 y x xe y x f y x . (a) Find the conditional density of X ,given y Y = , and that of Y , given . x X = (b) Find the density function of XY Z = . 5. An insurance company supposes that each person has an accident parameter and that the yearly number of accidents of someone whose accident parameter is λ is Poisson distributed with mean . They also suppose that the parameter value of a newly insured person can be assumed to be the value of a gamma random variable with parameter and . If a newly insured person has n accidents in her first year, find the conditional pdf of her accident parameter. Also, determine the expected number of accidents that she will have in the following year. 6. The number of customers using the automatic teller machine in a particular day follows a Poisson distribution with 180 = . The amount of money withdrawn by each customer is a random variable with mean $300 and standard deviation $500. (A negative withdrawal means that money was deposited.) Find the mean and variance of the total daily withdrawal. 7. Type i light bulbs function for a random amount of time having mean i μ and standard deviation i σ , . A light bulb randomly chosen from a bin of bulbs is a type 1 bulb with probability p , and a type 2 blub with probability 2 , 1 = i p 1.
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5 Assignment 5 (Due Date December 3, 2010) - 10/11 THE...

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