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 random variable taking the values \(0, 1, 2, \dots \), 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.

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.