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AMERICAN UNIVERSITY of BEIRUT STAT 230, Final Examination Time = 1 hour and 30 minutes Jan 25, 2008 You are allowed to use a calculator, one formula sheet, tables of discrete and continuous distributions. (GOOD LUCK!) 1. Consider the experiment of rolling a pair of fair 4-sided dice. Let X 1 be the outcome on die 1 , and X 2 be the outcome on die 2 respec- tively. Define Y 1 = max { X 1 , X 2 } and Y 2 = | X 1 - X 2 | . Determine the Covariance of Y 1 and Y 2 . [15 pts] 2. Let the joint pdf of the two random variables X and Y be f ( x, y ) = kx if 0 < x < y < 1, 0 elsewhere, and k is a positive constant. (a) Find the value of k such that f ( x, y ) is a joint pdf . [5 pts] (b) Determine the marginal pdf of X , f 1 ( x ), and the marginal pdf of Y , f 2 ( y ), respectively. [10 pts] (c) Are the random variables X and Y independent? [5 pts] 3. Let X 1 and X 2 be a random sample of size 2 from the standard normal distribution. Let Y 1 = ( X 1 + X 2 ) / 2 and Y 2 = ( X 1 - X 2 ) / 2. Show that Y 1 and Y 2 are independent standard normal random variables. Hint: Try to find the joint mgf of ( Y 1 , Y 2 ) = E ( e { t 1 Y 1 + t 2 Y 2 } ). [15 pts] 4. Let X 1 and X 2 be a random sample from the exponential distribution with common pdf f ( x ) = e - x if x > 0. Let Y 1 = X 1 /X

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