# Thread: how to show their sum is another distribution

1. ## how to show their sum is another distribution

Xi idd exponential distribution, show sum of n Xi is gamma distribution.

i think this is typical question. but i got no idea at all. So you try to show p.d.f.? or other ways?

2. Originally Posted by pengchao1024
give me a hint guys:

Xi iid some known distribution, how to show their sum is another distribution
i just want a hint on what to do to show that.
More details would help.

3. Use the product of the MGFs.
The sum of INDEPENDENT gamma's with the same BETA

So if $X_i\sim\Gamma (\alpha_i, \beta)$ then

$\sum_{i=1}^nX_i\sim\Gamma (\sum_{i=1}^n\alpha_i, \beta)$

That's also how you prove the sum of independent chi-squares is a $\chi^2$.

BUT you need independence.

I'm working on a random stopping boot strap problem where the chi-squares are NOT independent.
Anyone want to help me? lol

4. ## ok

how can u find the density function of Gamma distribution by using convolution?

5. Originally Posted by pengchao1024
how can u find the density function of Gamma distribution by using convolution?

painfully, that's an n-fold integral

6. ## ok

well, would you do it like first two variables for example to show the ideas of using induction