Let X1, ..., and Xn be independent random variables having a common distribution function that is...

 

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Let X1, ..., and Xn be independent random variables having a common distribution function that is specified up to an unknown parameter 8. Let T = T(X) be a function of the data X = (X1, ..., Xn). If the conditional distribution of X1, ..., Xn given T(X) does not depend on 8 then T(X) is said to be a sufficient statistic for 8. The following cases show that T(X) = Σ=1X is a sufficient statistic for 8. (a) They are normal with mean and variance 1. (b) The density of Xi is f(x) = 8e-0xx > 0. (c) The mass function of Xi is p(x) = x(1-0)1-x, x = 0, 1,0 <0 < 1. (d) The Xi are Poisson random variables with mean 8. Let X1, ..., and Xn be independent random variables having a common distribution function that is specified up to an unknown parameter 8. Let T = T(X) be a function of the data X = (X1, ..., Xn). If the conditional distribution of X1, ..., Xn given T(X) does not depend on 8 then T(X) is said to be a sufficient statistic for 8. The following cases show that T(X) = Σ=1X is a sufficient statistic for 8. (a) They are normal with mean and variance 1. (b) The density of Xi is f(x) = 8e-0xx > 0. (c) The mass function of Xi is p(x) = x(1-0)1-x, x = 0, 1,0 <0 < 1. (d) The Xi are Poisson random variables with mean 8.

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Solution The angular frequency w2rv 100... View the full answer