18. Let X1, . . . ,Xn be independent random variables having a common distribution function that is specified up to an unknown parameter θ. 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 θ then T(X) is said to be a sufficient statistic for θ. In the following cases, show that T(X) = ni

=1Xi is a sufficient statistic for θ.

(a) The Xi are normal with mean θ and variance 1.

(b) The density of Xi is f (x) = θe−θx, x > 0.

(c) The mass function of Xi is p(x) = θx(1 − θ)1−x, x = 0, 1, 0 < θ < 1.

(d) The Xi are Poisson random variables with mean θ.