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STAT-230_Kobeissi_Final_Exam_Fall_08-09

# STAT-230_Kobeissi_Final_Exam_Fall_08-09 - Exercise 4 Let X...

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American University of Beirut STAT 230 Introduction to Probability and Random Variables Fall 2008-2009 Final Exam Exercise 1 (6 points) Let f ( x, y ) = 2 , 0 x y < 1, be the joint pdf of X and Y . Find P ( X > Y 2 ). Exercise 2 (6 points) Let X 1 , X 2 , X 3 be independent random variables that represent lifetimes (in hours) of three key components of a device. Say their respective distributions are exponential with means 1000 , 1500, and 2000. Let Y = min( X 1 , X 2 , X 3 ). Find P ( Y > 1000). Exercise 3 (8 points) A consumer buys n light bulbs, each of which has a lifetime that has a normal distribution with mean 800 hours, and a standard deviation of 100 hours. A light bulb is replaced by another as soon as it burns out. Assuming independence of the lifetimes, find the smallest n so that the succession of light bulbs produces light for at least 10000 hours with probability 0
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Unformatted text preview: Exercise 4 Let X and Y be a couple of random variables with pdf f ( x,y ) = 1 x 2 y 2 x ≥ 1 , y ≥ 1 a. (6 points) ﬁnd the joint pdf of U = XY and V = X/Y b. (8 points) ﬁnd the marginal pdf of U and V Exercise 5 (8 points) Let X have a beta distribution with parameters α and β ; the pdf of X is given by: f ( x ) = Γ( α + β ) Γ( α )Γ( β ) x α-1 (1-x ) β-1 < x < 1 Show that E ( X ) = α α + β and V ar ( X ) = αβ ( α + β + 1)( α + β ) 2 Exercise 6 (8 points) Fifty numbers are rounded oﬀ to the nearest integer and then summed. If the individual round-oﬀ errors are uniformly distributed over the interval (-1 / 2 , 1 / 2). Find the probability that the resultant sum diﬀers from the exact sum by more than 3. good luck...
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