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) = n i=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−θ x , x > 0.

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

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