Compound Poisson process

The expected value of a compound Poisson process can be calculated using a result known as Wald's equation as: :\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). Making similar use of the law of total variance, the variance can be calculated as: : \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} Lastly, using the law of total probability, the moment generating function can be given as follows: :\Pr(Y(t)=i) = \sum_n \Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n) : \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} ==Exponentiation of measures==

Let N, Y, and D be as above. Let μ be the probability measure according to which D is distributed, i.e. :\mu(A) = \Pr(D \in A).\, Let δ0 be the trivial probability distribution putting all of the mass at zero. Then the probability distribution of Y(t) is the measure :\exp(\lambda t(\mu - \delta_0))\, where the exponential exp(ν) of a finite measure ν on Borel subsets of the real line is defined by :\exp(\nu) = \sum_{n=0}^\infty {\nu^{*n} \over n!} and : \nu^{*n} = \underbrace{\nu * \cdots *\nu}_{n \text{ factors}} is a convolution of measures, and the series converges weakly. ==See also==