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?

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- Oct 12th 2009, 02:00 PMpengchao1024how 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? - Oct 12th 2009, 02:57 PMmr fantastic
- Oct 12th 2009, 06:03 PMmatheagle
Use the product of the MGFs.

The sum of INDEPENDENT gamma's with the same BETA

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

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

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

You just add the dfs.

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 - Oct 13th 2009, 08:09 AMpengchao1024ok
how can u find the density function of Gamma distribution by using convolution?

- Oct 13th 2009, 08:11 AMmatheagle
- Oct 13th 2009, 08:12 AMpengchao1024ok
well, would you do it like first two variables for example to show the ideas of using induction