STAT 230 Final, Fall 2002

AMERICAN UNIVERSITY of BEIRUT STAT 230, Final Examination Time = 1 hour and 50 minutes Jan. 23, 2003 You are allowed to use one formula sheet, a table of discrete and continuous distributions, and a calculator! 1. Let A and B be two events such that P(4|B) = P(A|B’). Are the events A and B independent? (5 pts) . Let the pdf, f(z), be positive at z= —1,0,1 and zero elsewhere. (a) If f(0) = 1/4, find B(X?). (5 pts) (b) If f(0) = 1/4 and if E(X) = 1/4, what is Mx(¢), the moment generating function, of X? (5 pts) . Can you find a random variable X which has a Poisson distribution such that P(X = 0) =2P(X =1) =8P(X = 2)? (10 pts) . Consider an ordinary deck of 52 playing cards. Cards are drawn suc- cessively at random and one at a time. Find the probability of getting the fourth spade on the sixth draw if the sampling is done (a) with replacement. (5 pts) (b) without replacement. (5 pts) . Let X have a uniform distribution over the interval (—1,1). Define the transformation Y = In(x4;)*. (a) Find the pdf of Y. (10 pts) (b) Determine the mean and variance of ¥’. (5 pts) . Let the pdf f(z) = 1/2 if0 <2 < land 2< 2 < 3. Find the distribution function F(z) = P(X < xz). (10 pts) . Let f(z) = 4y if-o<2r<m. l+a? (a) For what value of the constant k f(z) is a probability density function (pdf)? (5 pts) (b) Show that the mean value does not exist. (5 pts)s