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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 Spring '10
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 Statistics, Probability

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