STAT 230 Final, Fall 2004

AMERICAN UNIVERSITY OF BEIRUT STATISTICS 230, Final Exam Jan 31, 2005 Time = 1 Hour and 30 Minutes You are allowed to use a formula sheet. 1. Let X be a discrete random variable that assumes positive probabilities on the set {1,2,---}. Show that B(X) = D%p1 — Fx(x)], when it exists. _ Let X and Y be two random variables with a joint cumulative dis- tribution function F(z,y) = P(X < 2,Y < y). Show that Fyx(x) + Fy(y) -1< F(z,y) < J Fx(x)Fyly), where Fx(z) and Fy(y) are the respective cumulative distribution functions of the random variables X and Y. . Let Xy,X2,°::,Xn be n independent and identically distributed ran- dom variables with common probability density function f(z) = ke7l2-4l if -co <Z< oO. (a) Determine the value of & such that f(x) is a probability density function. (b) Find 6, the maximum likelihood estimate of 8. Hint: You may need to minimize S77 |X; — @| (c) if n = 3, is the estimator found in (2) unbiased? . Let X denote the time required to do a computation using algorithm written in programming language A, and let Y denote the time required to the same calculations using programming language B. Assume fur- ther that X is normally distributed and with mean 10 seconds and standard deviation 3 seconds and Y is normally distributed with mean 9 seconds and standard deviation of 4 seconds. (a) What is the distribution of X — Y? (b) Find the probability that a given calculation will run faster using A than when using B.