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

−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.