Let a probability process follows the Poisson distribution with parameter ($\lambda$). Now each time an event has occurred, I will be paid $K \$$. What is the expected sum I will receive?
Actually, I am confused about probability mass function in: https://en.wikipedia.org/wiki/Poisson_distribution
The $\textit{pmf}$ of the poisson distribution talks about probability of occurrance of $n$ events. But I know, in order to calculate the expected sum, I need to calculate: $\sum_{k=1}^{\infty} k p(k)$. Where, $p(k)$ is the probability of occurrance of $k^{th}$ event.
Should I calculate conditional probability of each event, and thereby calculate the infinite sum?
I suspect you must find the expected amount of money that you will have received at moment $t$.
Let $N_t$ denote the number of events that occurred during time span $[0,t]$.
Then $N_t\sim\mathsf{Poisson}(\lambda t)$ so that $\mathbb EN_t=\lambda t$, and at moment $t$ you have receive $KN_t$ dollars.
We find the expectation as follows: $$\mathbb EKN_t=K\mathbb EN_t=K\lambda t$$
edit:
If you receive $k$ dollars at the $k$-th event then the amount of money received at moment $t$ is $$1+2+\cdots+N_t=\frac12N_t(N_t+1)$$
Can you find the expectation of this yourself?