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Unformatted text preview: P when X is an exponential random variable with parameter λ , and a = αλ , where 0 < α < 1. Problem 4: (Problem 74, page 95) Let X 1 , X 2 , ... be a sequence of idependent, identically distributed continuous random variables. We say that a record occurs at time n if X n > max( X 1 , ..., X n1 ). That is, X n is a record if it is larger than each of X 1 , ..., X n1 . Show 1. P { a record occurs at time n } = 1 n . 2. E [number of records by time n ] = ∑ n i =1 1 i . 3. V ar (number of records by time n ) = ∑ n i =1 i1 i 2 . 4. Let N = min { n : n > 1 and a record occurs at time n } . Show E [ N ] = ∞ . Problem 5: (Problem 76, page 96) Let X and Y be independent random variables with means μ X and μ Y and variances σ 2 X and σ 2 Y . Show that V ar ( XY ) = σ 2 X σ 2 Y + μ 2 Y σ 2 X + μ 2 X σ 2 Y ....
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This note was uploaded on 04/15/2010 for the course STAT 416 taught by Professor Denker during the Spring '08 term at Penn State.
 Spring '08
 DENKER
 Probability, Stochastic Modeling

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