The positive random variables $X_{1}, X_{2},...X_{n}$ are independent observations having the Gamma distribution $Ga(3,\frac{1}{\eta})$, with density function:
$\frac{x^{2}}{2\eta^{3}}e^{\frac{-x}{\eta}}$ $(x>0)$
Which depends on the unknown parameter $\eta \in (0,\infty)$.
(i) Find the maximum likelihood estimator of $\eta$.
My solution:
$\hat \eta = \frac{\bar X}{3}$
(ii) Show that the maximum likelihood estimator of $\eta$ is unbiased and find its variance.
Please may someone explain (or give me a hint) how to prove the estimator is unbiased?
An estimator is unbiased if its expectation equals the parameter; in this case, if $$\operatorname{E}[\hat \eta] = \operatorname{E}[\bar X/3] = \operatorname{E}[\bar X]/3 = \eta,$$ then $\hat \eta$ is unbiased for $\eta$.
Do you know how to determine $\operatorname{E}[X]$, the expectation of a single observation? If so, then use the linearity of expectation to compute $$\operatorname{E}[\bar X] = \operatorname{E}[(X_1 + X_2 + \cdots + X_n)/n].$$