Compound Poisson Process (CPP)

1. Definition: 

Remove the restriction that two or more customers cannot arrive at the same time, (i.e., remove orderliness property)

Let $N(t)$ be a Poisson Process with rate $\lambda$, and let $Y_i$ be the i.i.d random variable, then $X(t) = \sum^{N(t)}_{i=1} Y_i$ is a compound Poisson process

Example #1: Buses arrive according to a Poisson process. Let $Y_i$ be the number of people on bus i, and let $X(t)$ be the total number of people arriving by time t.

Example #2: Insurance claims arrive according to a Poisson process. Let $Y_i$ be the size of the claim (in dollars), and let $X(t)$ be the total amount due to all claims by time t.

2. Expectation: 

$E[X(t)] = E[E[X(t)|N(t)]]$

Since $E[X(t)|N(t) = n] = E[\sum^n_{i=1} Y_i] = nE[Y_i]$, i.e., $E[X(t)|N(t)]  = N(t)E[Y_i]$

So $E[E[X(t)|N(t)]] = E[N(t)E[Y_i]] = E[N(t)]*E[Y_i] = \lambda t E[Y_i]$

3. Variance: 

var[X(t)] = var[E[X(t)|N(t)]] + E[var[X(t)|N(t)]]

and we have $Var[X(t)|N(t)=n] = var(\sum^n_{i=1} E[Y_i]) = nVar[Y_i]$
or $var(X(t)|N(t)) = N(t)Var(Y_i)$.
So $Var[E[X(t)|N(t)]] + E[Var[X(t)|N(t)]] $
  = $Var[N(t)E(Y_i)] + E[N(t)Var[Y_i)]$
  = $ \lambda t E^2[Y_i] + \lambda t Var(Y_i)$
  = $\lambda t E^2[Y_i] + \lambda t (E[(Y_i)^2] - E^2[Y_i])$
  = $\lambda t E[(Y_i)^2]$

4. Example: