Compound Poisson process

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

where, ${\displaystyle \{N(t):t\ge 0\}}$ is the counting variable of a Poisson process with rate ${\displaystyle \lambda }$, and ${\displaystyle \{D_i:i\ge 1\}}$ are independent and identically distributed random variables, with distribution function G, which are also independent of ${\displaystyle \{N(t):t\ge 0\}.}$

When ${\displaystyle D_i}$ are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process.

Properties of the compound Poisson process

The expected value of a compound Poisson process can be calculated using a result known as Wald's equation as:

${\displaystyle \mathrm{E}(Y(t))=\mathrm{E}(D_1+\cdots +D_{N(t)})=\mathrm{E}(N(t))\mathrm{E}(D_1)=\mathrm{E}(N(t))\mathrm{E}(D)=\lambda t\mathrm{E}(D).}$

Making similar use of the law of total variance, the variance can be calculated as:

${\displaystyle \begin{array}{cc}\mathrm{var}(Y(t))& =\mathrm{E}(\mathrm{var}(Y(t)\mid N(t)))+\mathrm{var}(\mathrm{E}(Y(t)\mid N(t)))\\ & =\mathrm{E}(N(t)\mathrm{var}(D))+\mathrm{var}(N(t)\mathrm{E}(D))\\ & =\mathrm{var}(D)\mathrm{E}(N(t))+\mathrm{E}(D{)}^2\mathrm{var}(N(t))\\ & =\mathrm{var}(D)\lambda t+\mathrm{E}(D{)}^2\lambda t\\ & =\lambda t(\mathrm{var}(D)+\mathrm{E}(D{)}^2)\\ & =\lambda t\mathrm{E}(D^2).\end{array}}$

Lastly, using the law of total probability, the moment generating function can be given as follows:

${\displaystyle Pr(Y(t)=i)=\sum _{n}Pr(Y(t)=i\mid N(t)=n)Pr(N(t)=n)}$

${\displaystyle \begin{array}{cc}\mathrm{E}(e^{sY})& =\sum _i{e}^{si}Pr(Y(t)=i)\\ & =\sum _i{e}^{si}\sum _{n}Pr(Y(t)=i\mid N(t)=n)Pr(N(t)=n)\\ & =\sum _{n}Pr(N(t)=n)\sum _i{e}^{si}Pr(Y(t)=i\mid N(t)=n)\\ & =\sum _{n}Pr(N(t)=n)\sum _i{e}^{si}Pr(D_1+D_2+\cdots +D_n=i)\\ & =\sum _{n}Pr(N(t)=n)M_D(s{)}^n\\ & =\sum _{n}Pr(N(t)=n)e^{n\ln (M_D(s))}\\ & =M_{N(t)}(\ln (M_D(s)))\\ & =e^{\lambda t\left(M_D(s)-1\right)}.\end{array}}$