Does exponential distribution affect my solution?

Question

I was struck at the beginning that it was dealing with the exponential distribution, but as I got the answers, it was nothing to do with the exponential distribution. so I can just ignore the "exponential" right?

Answer 1

Multipart comment (may help you make sense of the problem).

(1) As @stiartstevemson says, $\bar X$ is an unbiased estimator for exponential mean $\beta$ because $$(\hat \beta) = E(\bar X) = \beta.$ The bias is 0 as you have seem.

(2) MSE = $E[(\hat \beta - \beta)^2] = Var(\hat \beta) + [b(\hat\beta)]^2.$ But have shown that the bias $b(\hat \theta) = 0,$ so the MSE is just the variance.

(I don't know why you conclude that you can/should ignore that you are dealing with an exponential distribution. Your computations may seem familiar and trivial, but for some other distributions such derivations get messy. So the assumption that the data are exponential is important.)