Let {N(t), t = 0} be a non homogeneous Poisson process with mean value function m(t). Given N(t) = n, then the unordered set of arrival times has the same distribution as n independent and identically distributed random variables having distribution function :
= $\frac{m(x)}{m(t)}$ for x<= t.
Suppose that workers incur accidents in accordance with a nonho- mogeneous Poisson process with mean value function m(t). Suppose further that each injured person is out of work for a random amount of time having distribution F. Let X(t) be the number of workers who are out of work at time t. Compute E[X(t)] and Var(X(t)).
Please Review
$\left(X(t) \, \middle| \, N(t)\right)$ can be written as a sum of indicator random variables:
Let $(Y_i)$ represent the indicator variable for the $(i)$-th injured person. Then we have:
$ Y_{i} = \begin{cases} 1 & \text{if the injured person is still out of the system}\\ 0 & \text{if the injured person is in the system.} \end{cases} $
$ \left(X(t) \, \middle| \, N(t)\right) = \sum_{i=1}^{n} Y_i $
After conditioning on arrival times of the accidents we take the expectation of both sides:
$ E\left[\left(X(t) \, \middle| \, N(t)\right)\right] = E\left[\left(\sum_{i=1}^{n} Y_i \, \middle| s_{1}, s_{2},\dots \right)P\left(s_{1},s_{2}, \dots \, \middle| \,N(t)=n \right) \,\right] $
$ E\left[\left(X(t) \, \middle| \, N(t)\right)\right] = n*\int_{0}^{t} (1-F(t-s))*(\frac{m(s)}{m(t)}) \,ds $
Here is a suggestion. Let $\Big\{[t_{j-1},t_{j})\Big\}_{j=1}^{n}$ be a partition of $[0,t)$ such that $t_0=0$ and $t_n=t$. Take $X_j$ as the number of accidents that occur on $[t_{j-1},t_j)$. Evidently $X_j\sim \text{Poisson}\left(m(t_j)-m(t_{j-1})\right)$.
If $n$ is very large we may regard the Poisson process on $[0,t)$ as one that splits the injured workers into two categories: those who are have returned to work by time $t$ and those who haven't.
It's not too difficult to see that the number of workers who were injured during $[t_{j-1},t_j)$ who have not returned to work by time $t$ is approximately $\text{Poisson}\Big((m(t_j)-m(t_{j-1}))(1-F(t-t_{j-1}))\Big)$.
By summing up and taking limits, we see that the distribution of the total numbers of worker who were injured on $[0,t)$ and haven't returned by time $t$ is $\text{Poisson}\Big(\int_0^tm'(x)(1-F(t-x))\mathrm{d}x\Big)$
Can you finish?