Unformatted text preview: P when X is an exponential random variable with parameter Î» , and a = Î±Î» , where 0 < Î± < 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 ] = âˆ‘ n i =1 1 i . 3. V ar (number of records by time n ) = âˆ‘ n i =1 i1 i 2 . 4. Let N = min { n : n > 1 and a record occurs at time n } . Show E [ N ] = âˆž . Problem 5: (Problem 76, page 96) Let X and Y be independent random variables with means Î¼ X and Î¼ Y and variances Ïƒ 2 X and Ïƒ 2 Y . Show that V ar ( XY ) = Ïƒ 2 X Ïƒ 2 Y + Î¼ 2 Y Ïƒ 2 X + Î¼ 2 X Ïƒ 2 Y ....
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 Spring '08
 DENKER
 Probability, Stochastic Modeling, Variance, Probability theory, exponential random variable, Geometric random variables

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