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Unformatted text preview: STAT 409 Homework #1 Fall 2009 (due Friday, September 4, by 4:00 p.m.) 1. Let > 0 and let X be a random variable with the probability density function f ( x ) = 1 + x , x > 1, zero otherwise. Let W = ln ( X ). What is the probability distribution of W ? 2. Let X be a Uniform ( 0, 1 ) and Y be a Uniform ( 0, 3 ) independent random variables. Let W = X + Y. Find and sketch the p.d.f. of W. 3. Suppose X 1 , X 2 , , X n are independent random variables, and X i has Geometric distribution with probability of success p i , i = 1, 2, , n . Let Y = min X i . What is the probability distribution of Y? Hint: Consider P ( X > x ) for a Geometric ( p ) random variable. 4. Every year on August 28, Anytown Tigers and Someville Lions play a soccer game. It is always a high-scoring game, the number of goals scored follows a Poisson process with the average rate of one goal per 5 minutes. What is the probability that the fifth goal is scored during the last 15 minutes of the first half? probability that the fifth goal is scored during the last 15 minutes of the first half?...
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