Compound Poisson process

<h2 id="properties-of-the-compound-poisson-process">Properties of the compound Poisson process</h2>
The <a href="/facts/Expected_value/1XV0JKL8">expected value</a> of a compound Poisson process can be calculated using a result known as <a href="/facts/Wald%27s_equation/hLuI7xHH">Wald's equation</a> as:

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    {\displaystyle \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 <a href="/facts/Law_of_total_variance/hDEf2nqK">law of total variance</a>, the <a href="/facts/Variance/ULBJKXD1">variance</a> can be calculated as:

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    {\displaystyle {\begin{aligned}\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{aligned}}}

Lastly, using the <a href="/facts/Law_of_total_probability/rq6NTRn1">law of total probability</a>, the <a href="/facts/Moment_generating_function/eTmk43QU">moment generating function</a> can be given as follows:

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

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    {\displaystyle {\begin{aligned}\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{aligned}}}

<h2 id="exponentiation-of-measures">Exponentiation of measures</h2>
Let N, Y, and D be as above. Let μ be the probability measure according to which D is distributed, i.e.

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    {\displaystyle \mu (A)=\Pr(D\in A).\,}

Let δ0 be the trivial probability distribution putting all of the mass at zero. Then the <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> of Y(t) is the measure

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    {\displaystyle \exp(\lambda t(\mu -\delta _{0}))\,}

where the exponential exp(ν) of a finite measure ν on <a href="/facts/Borel_set/jvSsfpkP">Borel subsets</a> of the <a href="/facts/Real_number/R02gw5Pb">real line</a> is defined by

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    {\displaystyle \exp(\nu )=\sum _{n=0}^{\infty }{\nu ^{*n} \over n!}}

and

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    {\displaystyle \nu ^{*n}=\underbrace {\nu *\cdots *\nu } _{n{\text{ factors}}}}

is a <a href="/facts/Convolution/PVPrdz9J">convolution</a> of measures, and the series converges <a href="/facts/Convergence_of_random_variables/pWdatFlY">weakly</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Poisson_process/8uphjAUc">Poisson process</a></li>
<li><a href="/facts/Poisson_distribution/CvCzXkHr">Poisson distribution</a></li>
<li><a href="/facts/Compound_Poisson_distribution/tLHJbhIP">Compound Poisson distribution</a></li>
<li><a href="/facts/Non-homogeneous_Poisson_process/8uphjAUc">Non-homogeneous Poisson process</a></li>
<li><a href="/facts/Campbell%27s_formula/LppwuXw1">Campbell's formula</a> for the <a href="/facts/Moment_generating_function/eTmk43QU">moment generating function</a> of a compound Poisson process</li></ul>

Compound Poisson process open-in-new

Compound Poisson process