Suppose that Y_{1}, Y_{2}, .... Y_{n}~exp(B). Define two estimators for B as:

B_{1}= nY_{(1)}; B_{2}= 1/n Summation from i-n of Y, or Y-bar.

I'm trying to determine that the first estimator is unbiased, but I've returned expected values of it as n^{3}B.

_{ Y(1) =}n[1 -Fy]^{n-1}fy,

= n (e^{-y/B})^{n-1}(1/B)(e^{-y/B})

= n/B (e^{-y/B})^{n}

= (n^{2}/B)(e^{-y/B})

E[B_{1}] = E[n* (n^{2}/B)(e^{-y/B})] = n^{3}E[(1/B)(e^{-y/B})] = n^{3}B.

Obviously this is incorrect. Where am I misunderstanding something? I've gone over the minimum order statistic function like ten times and the book we are using just glosses over the expectation of it.

Edit: I've noticed at least one mistake, putting (e^(-y/B))^n as n(e^(-y/B)) is not correct algebra. but, I'm still stuck.