Verify that theta hatnbspequiv x1n is an unbiased estimator

Question

Suppose that Y1 ,..., Yn are independent and identically distributed Exp(θ) random variables 

Let X1 ≡ Y1 + ... + Yn Use moment generating functions to show that X1 has a Gamma( n,θ ) distribution.

Let X2 ≡ mine (Y1,...,Yn) Show that X2 has an Exp(θ/n) distribution

Verify that theta hat ≡ X1/n is an unbiased estimator for theta

Verify that theta hat ≡ nX2 is an unbiased estimator for theta

Which estimator theta hat 1 or theta hat 2 is preferred estimation of theta