HW-Sol-5-V1 - MIT

Statistics for Engineers and Scientists Apr 11. MIT, Spring 2006 Handout #13. Solution 5. Problem 1: Let a > 0 be a known constant, and let > 0 be a parameter. Suppose X1 , X2 , .. , Xn is a sample from a population with one of the following densities. (a) The beta, ( , 1), density: fX (x | ) = x 1 , for 0 < x < 1. a (b) The Weilbull density: fX (x | ) = axa 1 e x , for x > 0. a . (c) The Pareto density: fX (x | ) = x( +1). , for x > a. In each case, nd a real-valued su cient statistic for . Solution Let X (X1 , X2 , .. , Xn ) be a collection of random variables Xi 's, and let x (x1 , x2.)

tial family of distribution. Recall that Gaussian distribution is a member of the exponential family of distribution and that random variables, X i’s and Y j’s, are mutually independent. Thus, their joint pdf belongs to the exponential family as well.

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  Distribution, Gaussian, Gaussian distribution

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