Show the statistics are unbiased for function $\frac{\alpha}{\lambda}$

Question

TASK:

Let $X_1,...,X_n$ a sample of i.i.d random variables from gamma distribution $(\lambda, \alpha)$ with parameter $\theta = (\lambda, \alpha)$. Show that statistics: $\frac{1}{n}\sum_{i=1}^{n}X_i$ are unbiased for parametric function $\frac{\alpha}{\lambda}$.

ANSWER: The mean for statistics is: $\lambda * \alpha$. But how can i show they are unbiased? What for there are parametric function?

Thank's for any help.

Answer 1

If $X\sim G(\lambda,\alpha)$, then $\mathbb{E}[X]=\dfrac{\alpha}{\lambda}$.

As expectation is linear, $\mathbb{E}\left[\dfrac{1}{n}\displaystyle\sum_{i=1}^n X_i\right]=\dfrac{1}{n}\displaystyle\sum_{i=1}^n \mathbb{E}[X_i]\stackrel{\text{i.i.d}}{=}\dfrac{1}{n}\displaystyle\sum_{i=1}^n \mathbb{E}[X]=\dfrac{1}{n}\left(n\dfrac{\alpha}{\lambda}\right)=\dfrac{\alpha}{\lambda}$.