# MLE of the Gamma Distribution

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 $$\operatorname{E}[\bar X] = \operatorname{E}[(X_1 + X_2 + \cdots + X_n)/n].$$
• @ConnorSimmons When $X_1, X_2, \ldots, X_n$ are IID, recall the formula $$\operatorname{Var}[X_1 + X_2 + \cdots + X_n] = \operatorname{Var}[X_1] + \operatorname{Var}[X_2] + \cdots + \operatorname{Var}[X_n].$$ Commented Apr 17, 2022 at 21:58