Let XX, be independent random variables having a common distribution function that is specified up to an unknown parameter 0.

Let T=T(X) be a function of the data X = (X,..., X). If the conditional distribution of X...., X, given T(X) does not depend on then T(X) is said to be a sufficient statistic for 6.

In the following cases, show that T(X) EX is a sufficient statistic for 0.

(a) The X, are normal with mean and variance 1.

(b) The density of X, is f(x) = 0

e, x>0.

(c) The mass function of X, is p(x)=0(10), x = 0, 1, 0 < <1.

(d) The X, are Poisson random variables with mean 0.