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Final-Fall2007-08 - that the first to arrive has to wait...

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American University of Beirut STAT 230 Introduction to Probability and Random Variables Fall 2007-2008 Final Exam Name: ............................... ID #: ................................. Exercise 1 Let X 1 , X 2 , X 3 be three independent random variables with binomial distributions b (4 , 1 / 2) , b (6 , 1 / 3), and b (12 , 1 / 6) respectively. a. find P ( X 1 = 2 , X 2 = 2 , X 3 = 5) b. find E ( X 1 X 2 X 3 ) c. find the mean and the variance of Y = X 1 + X 2 Exercise 2 The joint pdf of two random variables X and Y is f ( x, y ) = kx 0 < x < y < 1 a. find the value of the constant k b. find the marginal pdf of X and Y . Are they independent? c. find P ( X + Y < 1 / 2) d. find E ( X 2 Y ) Exercise 3 A man and a woman decide to meet at a certain location. If each person indepen- dently arrives at a time uniformly distributed between 12 noon and 1 PM, find the probability
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Unformatted text preview: that the first to arrive has to wait no longer than 10 minutes. Exercise 4 Let X and Y be a random sample of from the exponential distribution with pdf f ( x ) = e-x < x < ∞ Let U = X/Y and V = X + Y . Find the joint pdf of the couple ( U,V ). Are U and V independent? Exercise 5 Let X 1 ,X 2 ,...,X n be a random sample of size n with uniform distribution over (0 , 1). Find the pdf of Y = max( X 1 ,X 2 ,...,X n ). Find E ( Y ) ,V ar ( Y ) and M Y ( t ), the moment generating function of Y . good luck...
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