statistics (Sufficiency, etc)

1) Let X1, X2,...,Xn be a random sample from

f(x| theta) = 1/theta * e^(-x/theta) , for x>0, where the parameter theta >0

a) show that T=X1+X2+...+Xn is a sufficient statistics
b) Let theta hat = E[X1 | T]. Derive an expression for theta hat as a function as T.


2) Let X1, X2,..., Xn be a random sample fromt he statistical model { U(-theta, theta): theta is an element in (0, infinity)} Show that the statistic T = (a= Min(X1,..,Xn) , b=Max (X1,..., Xn) ) is a sufficient statistic, but not a minimal sufficient statistics.

3) Let X1,..., Xn be a random sample from N(0, sigma^2), argue that

(X1^2 + X2^2) / (X1^2+ X2^2+...+Xn^2) is independent of X1^2+X2^2+...+Xn^2


4) show that 1) the beta family Beta (alpha, beta), where beta is known, and 2) the negative binomial NB (r, p), where r is known, are a one-parameter exponential families


Please explain in details.