I have a question regarding Poisson events with death.
Assume time is continuous $t\in[0,\infty)$. A person may die with intensity $\delta$. Conditional on being alive, he may achieve a reward with an independent Poisson process with intensity $\eta$.
Here are my questions and my preliminary thoughts.
- What's the probability of get AT LEAST $n$ rewards before death? I thought it should be $$ (\frac{\eta}{\eta+\delta})^n $$ since \frac{\eta}{\eta+\delta} is the probability of get one reward before death. And Poisson process's memory-less feature leads to my conclusion. But not sure whether it is correct.
- Second, during a period of time $T$, what's his probability of getting EXACTLY $n$ rewards? Should be $$ e^{-\delta T} \frac{e^{-\eta T} (\eta T)^n}{n!} $$ Correct? But I am confused about can I use a coherent approach to get the solutions for question 1 and 2?
- Assume he dies either due to a $\delta$ event or achieve $N$ rewards. Then given he is alive, what's the probability distribution of number of rewards $n \in\{ 0,1,2,3...N-1\}$ he has achieved?
Please let me know if there is anything unclear or wrong. Thanks!