# how to show their sum is another distribution

• October 12th 2009, 02:00 PM
pengchao1024
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?
• October 12th 2009, 02:57 PM
mr fantastic
Quote:

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.
• October 12th 2009, 06:03 PM
matheagle
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
• October 13th 2009, 08:09 AM
pengchao1024
ok
how can u find the density function of Gamma distribution by using convolution?
• October 13th 2009, 08:11 AM
matheagle
Quote:

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

painfully, that's an n-fold integral
• October 13th 2009, 08:12 AM
pengchao1024
ok
well, would you do it like first two variables for example to show the ideas of using induction