A question about iid observatins $(X_1, \cdots ,X)n)$, knowing that $f_X(x) = ve^-vx$ , with x>0 and v>0.

Question

How do I show that X also have gamma distribution with parameters $nv$ and $n$? I know about the relationship between exp and gamma distributions, but i don't know how to solve this.

Answer 1

I think you mean that you have observations $X_1, \dots, X_n$ from an exponential distribution with rate $v$. In that case, $\bar X$ has a gamma distribution with shape parameter $n$ and rate parameter $nv$.

You can use moment generating functions to show that $\sum_i Xi$ has a gamma distribution with shape parameter $n$. Then divide the sum by $n$ to get the desired rate parameter.

If you mean something else, please try explaining more clearly. It seems someone has tried to edit your question, and not entirely successfully.