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Final-Fall-2004-05 - AMERICAN UNIVERSITY OF BEIRUT...

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Unformatted text preview: 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 E(X) z 3:111 — FX(2:)], when it exists. 2. Let X and Y be two random variables with a joint cumulative dis- tribution function F(:1:,y) = P(X g crrY _<_ y). Show that FX(3:) + Fy(y) — 1 g F(x,y) S ‘IFX(3:)Fy(y), where FX(.’L‘) and Fy(y) are the respective cumulative distribution functions of the random variables X and Y. 3. Let X1, X2, - - - , Xn be n independent and identically distributed ran- dom variables with common probability density function f (at) = Ice—'3‘“, if —00 < a: < oo. Determine the value of k such that f (cc) is a probability density function. Find 9, the maximum iikelihood estimate of 6. Hint: You may need to minimize Z? |X,; — 8| if n = 3, is the estimator found in (2) unbiased? 4. Let X denote the time required to do a computation using algorithm written in programming language A, and iet 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. ...
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