Compound Binomial and Geometric Sums

Two Random Counts, Two Compound Laws

A compound sum adds a random number of independent severities. The count describes how many events occur; each severity describes the amount contributed by one event. This construction appears, for example, when a fixed number of opportunities can each produce a claim, or when events continue until a terminating success. The distribution of the total depends on both the count law and the severity law.

The earlier material on random sums gives the general conditioning framework. Here we specialize it to binomial and geometric counts, deriving the corresponding transforms and moments. We use independent, identically distributed severities \(X_1,X_2,\ldots\), independent of the count \(N\), and define \(S=0\) when \(N=0\).

The zero-based geometric convention is useful when each counted event occurs before an independent terminating event. Another common convention counts trials through and including the first success, giving support \(1,2,\ldots\). These are different counts: if \(N_0\) counts failures, the trials-including-success count is \(N_0+1\). In a compound sum, that extra trial would add an extra severity, so the convention must be stated.

For a general severity \(X\), let \(\phi_X(t)=E[e^{itX}]\) denote its characteristic function. When \(X\) is nonnegative and integer-valued, let \(G_X(z)=E[z^X]\), for \(|z|\le1\), denote its probability-generating function. The probability-generating-function formulas below require this integer-valued condition; the characteristic-function formulas do not.

Transform Formulas

Condition on the count. Given \(N=k\), the total is a sum of \(k\) independent severities, so its characteristic function is \(\phi_X(t)^k\). Averaging this quantity over the count distribution produces the compound characteristic function.

To derive the binomial formula, first condition on \(N\):

The last equality is the binomial theorem. Replacing \(\phi_X(t)\) by \(G_X(z)\) gives the probability-generating-function formula: given \(N=k\), independence gives \(E[z^S\mid N=k]=G_X(z)^k\), and the same finite sum applies.

For the geometric count, the corresponding sum is an infinite series. It converges for every real \(t\), since \(|q\phi_X(t)|\le q<1\). For the probability-generating function and \(|z|\le1\), \(|qG_X(z)|\le q<1\) as well. Thus the geometric series formula applies in each case:

The same calculation with \(G_X(z)\) in place of \(\phi_X(t)\) proves the geometric probability-generating-function formula. The denominator is nonzero because the magnitude of the subtracted term is at most \(q<1\).

Means and Variances

Suppose \(E[X_i]=\mu\) and \(\operatorname{Var}(X_i)=\sigma^2<\infty\). Given \(N\), the conditional mean of the sum is \(N\mu\), and its conditional variance is \(N\sigma^2\). The law of total expectation and law of total variance therefore give \(E[S]=E[N]\mu\) and \(\operatorname{Var}(S)=E[N]\sigma^2+\operatorname{Var}(N)\mu^2\). For these uses, independence of \(N\) and the iid severities is essential.

The binomial count has \(E[N]=mp\) and \(\operatorname{Var}(N)=mp(1-p)\). For the zero-based geometric count, \(E[N]=q/r\) and \(\operatorname{Var}(N)=q/r^2\). Substituting yields:

For completeness, the geometric count moments follow directly from the probability-generating function \(G_N(z)=r/(1-qz)\). Differentiation gives \(G_N'(z)=rq/(1-qz)^2\) and \(G_N''(z)=2rq^2/(1-qz)^3\). Hence \(E[N]=G_N'(1)=q/r\) and \(E[N(N-1)]=G_N''(1)=2q^2/r^2\). Since \(N^2=N(N-1)+N\),

To verify the random-sum variance formula, condition on \(N\). The law of total variance states \(\operatorname{Var}(S)=E[\operatorname{Var}(S\mid N)]+\operatorname{Var}(E[S\mid N])\). Conditional independence of the severities gives \(\operatorname{Var}(S\mid N)=N\sigma^2\), while \(E[S\mid N]=N\mu\). Therefore

The expectation formula follows in the same way from \(E[S]=E[E[S\mid N]]=E[N\mu]=E[N]\mu\). Substituting the respective count moments proves both sets of compound moment formulas.

Interpretation and Common Pitfalls

The two count models represent different event mechanisms. A binomial count is bounded by \(m\): there are \(m\) opportunities, each included with probability \(p\). Its transform is a finite power. A geometric count has no fixed upper bound: continuation occurs with probability \(q\) at each step until termination. Its transform is a convergent geometric series. In both cases, the count's variability contributes to the variance of the total through the term \(\operatorname{Var}(N)\mu^2\).

• Using the wrong geometric convention. A count of failures before success has mean \(q/r\), not \(1/r\); the latter is the mean number of trials through success. Decide what is being counted before applying a formula.
• Dropping count variability from the variance. The expression \(E[N]\sigma^2\) accounts only for severity variation at a fixed count. The additional \(\operatorname{Var}(N)\mu^2\) term accounts for variation in the number of summands.
• Using a probability-generating function for noninteger severities. The generating-function formulas require nonnegative integer-valued severities. For continuous or general real-valued severities, use the characteristic-function formulas instead.
• Assuming a zero total is impossible. Under the zero-based geometric convention, \(P(N=0)=r\), so the sum is zero whenever no events are counted. More generally, a zero total can also occur if included severities themselves can equal zero.