STAT230_Final_Summer08

# STAT230_Final_Summer08 - American University of Beirut STAT...

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American University of Beirut STAT 230 Introduction to Prob0ability and Random Variables Summer 2008 Final Exam Name: ............................... ID #: ................................. Section: 4 (7:30) 6 (12:30) Exercise 1 (10 points) The number of accidents in a period of one week follows a Poisson distribution with mean two. The number of accidents from week to week are independent. Find the probability of exactly seven accidents in a given three weeks. Exercise 2 Let X be a continuous random variable with pdf f ( x ) = kx 2 e - x 2 / 2 , 0 < x < + a. (10 points) find the value of the constant k b. (10 points) find E ( X ) and V ar ( X ) Exercise 3 (20 points) Let X 1 and X 2 be independent χ 2 (2) distributions, i.e. f ( x ) = 1 2 e - x/ 2 0 < x < + find the joint pdf of Y 1 = X 1 + X 2 and Y 2 = X 1 - X 2 , then find the marginal pdf of Y 2 Exercise 4 (20 points) Flip n = 8 fair coins and remove all that come up heads. Flip the other (tails) coins and remove the heads. Continue flipping the remaining coins until each has come up heads. We shall find the pdf of Y , the number of trials needed. Let X i , equal the number of flips required to observe heads on coin
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