# Thread: How to get distribution of X1/(X1+X2+...Xn)

1. ## How to get distribution of X1/(X1+X2+...Xn)

Hi Everyone,
I've got a problem on how to find the pdf and cdf for X1/(X1+X2+...Xn) , where xi is iid exponential(1, 1/s)?
I read something about that using the fact that x2+...Xn is gamma(n-1,s), but I don't know what to do next. I think it may be associated with transformation, but I don't know how to do it?

Anyone can provide some suggestion? Really appreciated.

2. ## Re: How to get distribution of X1/(X1+X2+...Xn)

It's been a while since I've done that problem but let's see... We did it for the general case of gamma distributions (which the exponential is a special case). I think we only had two gammas though (say X and Y). We can reduce your case down to this by realizing that the sum of exponentials is gamma and since are independent then and are independent. Then I think it's just a simple bivariate transformation to get the result you want. (Hint: you might be shooting for a beta...)

3. ## Re: How to get distribution of X1/(X1+X2+...Xn)

So let

Since the support of are ,
the support of is

One common way:
Since you know the pdf of the exponential distribution and the gamma
distribution, the joint pdf of and are known. Then you can consider
an auxillary random variable , and then you can determine the
joint pdf of and by taking the two-variables jacobian transformation.
After this, you integrate over all the support of to obtain
back the marginal pdf of

Another way:
Note

Then
by the independence of and

Differentiating both sides with respect to , we have

Subsitute all the known pdf inside with your parameterization, then
do the gamma integral.

4. ## Re: How to get distribution of X1/(X1+X2+...Xn)

Thank you for posting BMG. That second solution was what I was thinking of, but was too lazy to post it up.

5. ## Re: How to get distribution of X1/(X1+X2+...Xn)

Originally Posted by hslinhc
...where xi is iid exponential(1, 1/s)?
The exponential distribution only has one parameter. Did you mean Gamma(1,1/s)?

6. ## Re: How to get distribution of X1/(X1+X2+...Xn)

Thank you Dason.
And thank you BGM. Your second solution is cool.

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