I have the following question in one of my probability books and I have just picked up studying Joint PMFs but something doesn't seem to add up here.
Let $N$ and $K$ be random variables having the joint PMF:
$$P_{N,K}(n,k)=\begin{cases}
\frac{100^ne^{-100}}{(n+1)!} & k=0,1,2,…,n ; n=0,1,2,…\\
0 & \textrm{otherwise}
\end{cases}$$
Now since this is a joint PMF shouldn't running a double sum give 1? However when I tried running that sum on WolframAlpha I get a conditioned result for the convergence of the sum here
Can someone explain what's going wrong?
Best Answer
Here is what's going on: $$\sum_{n=0}^\infty\underbrace{\sum_{k=0}^n}_{=n+1}\frac{100^ne^{-100}}{(n+1)!}=\sum_{n=0}^\infty\frac{100^ne^{-100}}{n!}=e^{-100}e^{100}=1$$