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

Related

• Poisson Process
• Nonhomogeneous Poisson Process