0
$\begingroup$

I understand how to find the pdf for the sum of $N$ exponentially distributed random variables, but how do I find the pdf when $N$ is also an independent random variable.

Here is the problem:

Let $N,X_1,X_2,...$ be independent, where $P[N=n]=q^{n-1}p, n \ge 1$ and each $X_k$ has the exponential density $f(x;\alpha;1)$. Show that $X_1+..+X_N$ has density $f(x;\alpha p,1)$. This is problem 20.18 in Billingsley.

where

$f(x;\alpha,u)={\alpha^u\over\Gamma(u)} x^{u-1}e^{- \alpha x}$

I don't know where to start. I think this relates to the Erlang pdf, but I am not sure how to connect the two. Any help would be greatly appreciated.

$\endgroup$
1
$\begingroup$

For each $n\geqslant1$, let $g_n$ denote the PDF of $X_1+\cdots+X_n$, then, by independence of $N$ and the sequence $(X_n)$, the density $g$ of the random variable $X_1+\cdots+X_N$ is $$g=\sum_{n\geqslant 1}P(N=n)g_n=\sum_{n\geqslant 1}pq^{n-1}g_n,$$ hence, if every $g_n$ is known, one can compute $g$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.