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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 3 Fall 2007 Issued: Thursday, September 13 Due: Thursday, September 20 Problem 3.1 Consider an exponentially distributed variate with density f X ( t ) := braceleftBigg exp(- t ) for t > otherwise . The parameter > 0 is known as the failure rate. (a) Suppose X has an exponential distribution rate with failure rate . Find the distribu- tion of Y = X ; (b) Let X 1 ,... ,X n be a sample from an exponential distribution, and let X = X 1 + ... + Xn n . Show that X and ( X 2 1 + ... + X 2 n ) X 2 are independent. (c) Let X (1) X ( n ) be the order statistics and X = X 1 + + X n n the sample average. Show that X and X (1) X ( n ) are independent. Problem 3.2 Determine the cumulant generating function A and for the associated exponential family of dimension one with X = R , T ( x ) = x , and (a) h ( x ) = exp(- x 2 )....
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- Fall '08