Editing Compound Poisson process (section)

==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:

:<math>\operatorname E(Y(t)) = \operatorname E(D_1 + \cdots + D_{N(t)}) = \operatorname  E(N(t))\operatorname E(D_1) = \operatorname  E(N(t)) \operatorname E(D) = \lambda t \operatorname  E(D).</math>

Making similar use of the [[law of total variance]], the [[variance]] can be calculated as: 
:<math>
\begin{align}
\operatorname{var}(Y(t)) &= \operatorname  E(\operatorname{var}(Y(t)\mid N(t))) + \operatorname{var}(\operatorname E(Y(t)\mid N(t))) \\[5pt]
&= \operatorname E(N(t)\operatorname{var}(D)) + \operatorname{var}(N(t) \operatorname E(D)) \\[5pt]
&= \operatorname{var}(D) \operatorname E(N(t)) + \operatorname E(D)^2 \operatorname{var}(N(t)) \\[5pt]
&= \operatorname{var}(D)\lambda t + \operatorname E(D)^2\lambda t \\[5pt]
&= \lambda t(\operatorname{var}(D) + \operatorname E(D)^2) \\[5pt]
&= \lambda t \operatorname E(D^2).
\end{align}
</math>

Lastly, using the [[law of total probability]], the [[moment generating function]] can be given as follows:
 
:<math>\Pr(Y(t)=i) = \sum_n \Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n) </math>

:<math>
\begin{align}
\operatorname E(e^{sY}) & = \sum_i e^{si} \Pr(Y(t)=i) \\[5pt]
& = \sum_i e^{si} \sum_{n} \Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n) \\[5pt]
& = \sum_n \Pr(N(t)=n) \sum_i e^{si} \Pr(Y(t)=i\mid N(t)=n) \\[5pt]
& = \sum_n \Pr(N(t)=n) \sum_i e^{si}\Pr(D_1 + D_2 + \cdots + D_n=i) \\[5pt]
& = \sum_n \Pr(N(t)=n) M_D(s)^n \\[5pt]
& = \sum_n \Pr(N(t)=n) e^{n\ln(M_D(s))} \\[5pt]
& = M_{N(t)}(\ln(M_D(s))) \\[5pt]
& = e^{\lambda t \left( M_D(s) - 1 \right) }.
\end{align}
</math>