Disclaimer I'm pretty inexperienced at statistics so I apologize if this is a simple question that I have made too complicated!
Question Suppose we have the following situation: there are $N$ discrete bins $1,\dots,N$. At a particular time $t=0$, each bin has some number $n_k$ of particles in it ($k=1,\dots,N$). At each discrete time step, for each $k=1,\dots,N$, every particle in bin $k$ undergoes a Bernoulli trial with success probability $p_k$ (note that $p_k\neq{p_j}$ for $j\neq{k}$ in general). If the trial is successful, the particle moves to bin $k+1$ (if $k=N$, the particle leaves the system). If the trial is unsuccessful, the particle is said to expire, and also leaves the system.
Let $E$ denote the number of particles which expire (i.e. do not reach bin $N$ and have a successful Bernoulli trial in bin $N$). I have the following estimate for the expected value of $E$:
$$ \mathbb{E}\big[E\ \big]=\sum_{t=1}^N\big[n_t\big(1-\prod_{k=t}^Np_k\big)\big] $$
My reasoning is: the quantity $\prod_{k=t}^Np_k$ is the probability that a particle which has made it to bin $t$ will make it out of the last bin without expiring. We then negate this quantity to get the probability that it does expire, multiply by the number of particles in the bin $t$, then sum over all bins.
- Is the estimate for $\mathbb{E}\big[E\ \big]$ correct?
- What is the variance of $E$? How can I compute it? I'm not sure what $\mathbb{E}\big[E^2\big]$ would be.
- Is there a well-known distribution or something like that which describes this situation? It seems related to the Poisson-Binomial distribution, but I can't quite get my finger on it.
Thanks!