__1. Definition:__

__1. Definition:__

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

Let be a Poisson Process with rate , and let be the i.i.d random variable, then is a

**compound Poisson process**

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

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

__2. Expectation: __

__2. Expectation:__

Since , i.e.,

So

**3. Variance:**

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