Compound Poisson Process

Definition

A Compound Poisson Process is defined as:

$X(t)={\sum }_{i=1}^{N(t)}{Y}_i$

where:

• $\{N(t),t\ge 0\}$ is a Poisson process with rate $\lambda$
• $Y_1,Y_2,\dots$ are i.i.d. random variables, independent of $N(t)$

Interpretation

Events arrive according to a Poisson process, and each event carries a random “payload” $Y_i$. The compound process tracks the cumulative total. Example: insurance claims arrive as Poisson, each with a random claim amount $Y_i$; $X(t)$ is the total payout by time $t$.

Moments

$E[X(t)]=\lambda t\cdot E[Y_i]$

$\text{Var}(X(t))=\lambda t\cdot E[Y_i^2]$

Derived via conditional expectation: $E[X(t)]=E[E[X(t)\mid N(t)]]=E[N(t)\cdot E[Y]]=\lambda t\cdot E[Y]$.

Example

Scenario: Customers arrive at a store according to a Poisson process with rate $\lambda =5$ per hour. Each customer spends a random amount $Y_i$ with E[Y] = \20$and$E[Y^2] = 600$.

What $X(t)$ tracks: Total revenue by time $t$ (not customer count).

Process	Formula	Tracks
$N(t)$	Poisson count	Number of customers
$X(t)$	${\sum }_{i=1}^{N(t)}{Y}_i$	Total revenue

Expected revenue in 2 hours:

$E[X(2)]=\lambda t\cdot E[Y]=5\times 2\times 20=\$200$

Variance:

$\text{Var}(X(2))=\lambda t\cdot E[Y^2]=5\times 2\times 600=\$6000$

Relation to Poisson: When every customer spends exactly $1$ (i.e., $Y_i\equiv 1$), the compound process reduces to the regular Poisson process: $X(t)=N(t)$.

Related

• Poisson Process
• Nonhomogeneous Poisson Process