5 COMPOUND DISTRIBUTIONS

A compound distribution arises when a random number of random quantities are added together. The number of insurance claims in a year is random; so is the size of each claim; the total paid out is the sum of a random number of random terms. Write \[S = X_1 + X_2 + \cdots + X_N ,\] where \(N\) is a non-negative integer-valued random variable, the \(X_i\) are independent and identically distributed, and \(N\) is independent of the \(X_i\). The empty sum is zero, so \(S=0\) when \(N=0\).

Everything in this section follows from conditioning on \(N\), which is why it sits immediately after the previous chapter.

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