5.1 Mean, Variance and Generating Function
Theorem 5.1.1. For the compound sum \(S\) above, writing \(\mu = E(X_i)\) and \(\sigma ^{2} = \Var (X_i)\), \[E(S) = E(N)\,\mu ,\] \[\Var (S) = E(N)\,\sigma ^{2} + \Var (N)\,\mu ^{2},\] \[M_S(t) = G_N\!\left (M_X(t)\right ).\]
Proof.
The mean
Condition on \(N\) and use the tower property. Given \(N=n\) the sum has exactly \(n\) terms, so \[E\left (S \mid N=n\right ) = E\left (X_1+\cdots +X_n\right ) = n\mu , \qquad \text {that is}\qquad E\left (S\mid N\right ) = N\mu .\] Hence \[E(S) = E\left [E\left (S\mid N\right )\right ] = E(N\mu ) = E(N)\,\mu .\]
The variance
Given \(N=n\) the terms are independent, so their variances add: \[\Var \left (S \mid N\right ) = N\sigma ^{2}.\] Applying the conditional variance formula of the previous chapter, \begin {align*} \Var (S) &= E\left [\Var \left (S\mid N\right )\right ] + \Var \left [E\left (S\mid N\right )\right ]\\ &= E\left (N\sigma ^{2}\right ) + \Var \left (N\mu \right )\\ &= E(N)\,\sigma ^{2} + \Var (N)\,\mu ^{2}, \end {align*}
the constants \(\sigma ^{2}\) and \(\mu ^{2}\) coming out of the expectation and the variance respectively.
The moment generating function
Again conditioning on \(N\), and using that the \(X_i\) are independent with common moment generating function \(M_X\), \[E\left (e^{tS} \mid N=n\right ) = E\left (e^{t(X_1+\cdots +X_n)}\right ) = \left [M_X(t)\right ]^{n}.\] Therefore \[M_S(t) = E\left [E\left (e^{tS}\mid N\right )\right ] = E\left (\left [M_X(t)\right ]^{N}\right ) = G_N\!\left (M_X(t)\right ),\] the last step being the definition of the probability generating function of \(N\) evaluated at the number \(M_X(t)\). □
Note. The variance formula has two terms and both are needed. The first, \(E(N)\sigma ^{2}\), is the variability of the claim sizes; the second, \(\Var (N)\mu ^{2}\), is the variability in how many there are. Setting \(N\) equal to a constant \(n\) kills the second term and returns the familiar \(n\sigma ^{2}\). Forgetting it is the standard error here, and it understates the variance badly whenever \(N\) is itself highly variable.
Remark. The formula \(M_S = G_N \circ M_X\) deserves attention. It is a composition of generating functions, not a product — the product rule of the earlier chapter applies to a fixed number of summands, and here the number is random. The outer function is the probability generating function of the counting variable and the inner is the moment generating function of the terms, which is exactly what makes the units work: \(M_X(t)\) is a number, and \(G_N\) takes numbers.
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