Unformatted text preview: EX and Var X . Problem 4. Suppose n balls are distributed at random into r boxes. Let X i = 1 if the i th box is empty, X i = 0 otherwise. (a) Compute EX i . (b) Compute EX i X j , for i 6 = j . (c) Let S be the number of empty boxes. Find ES and Var S . Problem 5. Suppose ( X,Y ) is uniformly distributed on the 2dimensional unit disc D 2 = { ( x,y ) : x 2 + y 2 ≤ 1 } . Find . .. (a) . ..the marginal cdf F X of X . (b) . ..the conditional cdf of Y given X = 1 / 2. The following problem is optional for students enrolled in 360, required for students enrolled in 560. Problem 6. Suppose X is a continuous random variable with pdf f , and suppose that X is bounded. That is, there exists a constant a > 0 such that P [a ≤ X ≤ a ] = 1. Prove that the moment generating function φ X of X satisﬁes φ X ( t ) < ∞ for all real t . Hint: ﬁrst assume t > ....
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 Fall '07
 EHRLICHMAN
 Variance, Probability theory, probability density function, continuous random variable, moment generating function

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