the compound Poisson process

Often the events themselves carry a random size, and you care about the running total of those sizes, not just the count. An insurer cares about total claim money, not how many claims; a website cares about total bytes served, not how many requests. The compound Poisson process tracks exactly this: events arrive as a Poisson process, and each event brings along an independent random amount, which you keep adding up.

Formally, let N(t) be a Poisson process of rate lambda, and let Y1, Y2, Y3, ... be independent and identically distributed jump sizes, independent of N. The compound process is the random sum X(t) = Y1 + Y2 + ... + Y_{N(t)} — add up the sizes of all events that have happened by time t (and X(t) = 0 if none have). Both the number of terms AND the terms are random. Its mean is beautifully simple by conditioning: E[X(t)] = E[N(t)] * E[Y] = lambda*t*E[Y]. Its variance follows from the law of total variance and is Var(X(t)) = lambda*t*E[Y^2] (note: the SECOND moment of Y, not its variance — a common slip). So you scale up the per-event statistics by the expected number of events.

This is the standard model of accumulated, lumpy risk. In insurance it is the classical aggregate-claims model; in finance it appears in jump models of asset prices; in inventory it models total demand. Two honest cautions: X(t) is generally NOT Poisson-distributed (it is only Poisson when each Y is exactly 1, i.e. ordinary counting), and its distribution usually has no neat closed form — you reach for its characteristic function or moment generating function, or simulate it. The clean part is the moments, which the conditioning argument hands you immediately.