2
$\begingroup$

Let $X_1, X_2, \ldots$ independent and identically distributed discrete random variables with support contained in $\mathbb{N}$ and let $N=X_1+1$. How can I calculate the moment generating function of $$S_N=X_1+\cdots+X_N?$$

Thanks for any help.

My attepmt: We have that $$M_{S_N}(t)=E[e^{rS_N}]=E[e^{r0}]Pr(N=0)+\sum_{n=1}^\infty E[e^{r(X_1+\cdots+X_n)}|N=n]Pr(N=n) $$

$\endgroup$
  • $\begingroup$ If you provide some context and share your attempts on solving the problem you are more likely to get responses. $\endgroup$ – PierreCarre Mar 29 at 10:26
1
$\begingroup$

$Ee^{tS_N}=\sum_n EI_{X_1=n}e^{t(X_1+X_2+..+X_{n+1})}=\sum_n M(t)^{n}P(X_1=n)e^{tn}=\sum_n P(X_1=n)e^{(t+\log(M(t))n}=M(t+\log(M(t))$.

$\endgroup$

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.