The calls to the fire department occur according to a Poisson process with a rate of three per day. The fire department must respond to each call. Of the calls that come into the fire department, on the average one third turn out to be false alarms. Firemen are presently paid 140 Dollars a day. A new plan is proposed where they will be paid a random amount of money per fire that they actually fight. In the new scheme the expected pay per fire fought is 60 Dollars. Determine the expected reward until time t.
[Resnick, Adventures in Stochastic Processes]
My approach is the following: Let $T _n=\sum\nolimits_{i=1}^{N_ t}X_ i $ where $T _n$ is the time of the nth call and $X _i $ the time between the calls. The number of calls in $[0,t]$ is given as $N _t = \sum\nolimits_{n=1}^{\infty}1 _{(T_ n\leq t)}$. Let $Y _n$ the reward associated with $X_ n$. Then we have for the total reward at time t: $R _t = \sum\nolimits_{i=1}^{N _t}Y_ i $.
So what I have to do now is to compute the expected value of $R_t$ . How can I do that? Can I use Wald's identity: $E(R _t)=E(N _t)E(Y _i)$?
In the usual case, the rewards are independant from the fire arrival times. That's what implied in A new plan is proposed where they will be paid a random amount of money per fire that they actually fight.
Therefore you can assume that the rewards process $(Y_i)_{i}$ is independant from the time of the calls $(T_i)_i$ and $(N_t)_{t>0}$ as well, because it is a measurable function of $(T_i)_i$. Then you can apply your result (Wald's formula) which is easily derived using iterated expectation as follows :
$\mathop{\mathbb{E}}[R_t] = \mathop{\mathbb{E}}[\mathop{\mathbb{E}}[R_t|N_t]] = \mathop{\mathbb{E}}[\mathop{\mathbb{E}}[Y_1]N_t] = \mathop{\mathbb{E}}[Y_1]\mathop{\mathbb{E}}[N_t]$