Sum of a Random Number of Independent Random Variables

Aggregate insurance claims are a sum of individual claim sizes — but the number of claims is itself random. Aggregate trading P&L over a day is a sum of individual trade P&Ls, and the number of trades is random too. Every result in Derived Distributions and Transforms and Moment Generating Functions assumed a fixed number of terms in a sum. This note drops that assumption: $Y=X_1+\cdots+X_N$ where $N$ itself is a random variable, independent of the i.i.d. $X_i$'s. The law of iterated expectations and law of total variance turn out to be exactly the right tools — condition on $N=n$ (reducing to the familiar fixed-sum case), then average over $N$'s own randomness — and the transform machinery extends cleanly too, closing with two genuinely surprising closure results.

Learning objectives. By the end of this note you should be able to: