I already know that this is a Gamma Distribution but I believe I am getting the wrong parameters. I have used the following steps to arrive at the answer. (We are given n exponential random varaibles with parameter $\theta$ )
$\frac{ \sum_{i=1}^{n} Y_i }{n} = \overline{Y}$. Therefore $\sum_{i=1}^{n} Y_i = n\overline{Y}$
Therefore we compute the moment genertating function of $n \overline{Y}$ which is equal to $M_{n \overline{Y}}(t) = \prod_{i=1}^{n} M_{Y_{i}}(t) = \frac{1}{ (1- \theta t )^n}$.
This implies that $M_{\overline{Y}}(nt) = \frac{1}{ (1- \theta t )^n} $. Thus substituting $u=nt$ , we get $M_{\overline{Y}}(u)= \frac{1}{ (1- \frac{ \theta}{n} u )^n}$. Therefore the parameters are $(n,\frac{\theta}{n})$. But the other answers on stackexchange state that $(n,n \theta) $ are the parameters of the gamma distribution. What mistake have I made ? Thank you for the help!
Find the distribution of the average of exponential random variables -Was refering to this other stackexchange answer
