STAT 230 Exam 2, Fall 2007

American University of Beirut STAT 230 Introduction to Probability and Random Variables Fall 2007-2008 quiz # 2 Name: ............................... ID #: ................................. Exercise 1 (25 points) Let X be a random a random variable with pdf f(x) =      1/2 0 < x < 1 1/2 2 < x < 3 0 elsewhere a. find the cdf of X b. find P(1/3 < X ≤ 7/3) c. find E(X) and V ar(X) d. find MX(t), the moment generating function of X Exercise 2 (20 points) The life X in years of a voltage of a car has the pdf f(x) = 3x2 73 e−(x/7)3 0 < x < +∞ a. find the probability that the regulator will last at least 7 years. b. given that it has lasted at least 7 years, find the probability that it will last at least another 3.5 years. Exercise 3 (25 points) Let X be a random variable with pdf f(x) = e−x (1 + e−x)2 − ∞ < x < +∞ Show that Y = 1 1 + e−X have a uniform distribution U(0, 1) Exercise 4 (30 points) Let X be a random variable with pdf f(x) = a e−|x| − ∞ < x < +∞ a. find the value of the constant a b. show that E(Xn) = 1 2 (1 + (−1)n).n! c. let Z = |X|. Find the pdf of Z