Random Sums with Independent Counts

Definition A random count is a random variable $N$ whose possible values are nonnegative integers $0, 1, 2, \dots$. In a random sum it is the number of terms that are added. The results on this page assume that the count is independent of the whole sequence of terms $X_1, X_2, \dots$, which means that $N$ is independent of the vector $(X_1, \dots, X_k)$ for every $k$. Independence of $N$ from each term separately is weaker, and it is not enough for the variance formula below.

Definition Let $X_1, X_2, \dots$ be IID random variables and let $N$ be a random count independent of the whole sequence. The random sum is

so on an outcome where $N = n$ the value of $S$ is $X_1 + \dots + X_n$, the partial sum $S_n$ of the first $n$ terms. For each $n$ with $P(N = n) > 0$, independence makes the conditional distribution of $(X_1, \dots, X_n)$ given $N = n$ equal to its unconditional distribution. Given $N = n$, the random sum is therefore an ordinary sum of $n$ IID terms.

Definition A sum with no terms, such as $\sum_{i=1}^{0} X_i$, is defined to be $0$. So $S = 0$ on the event $\{N = 0\}$. This convention keeps the rule $S_{n+1} = S_n + X_{n+1}$ for the partial sums true at $n = 0$, and it makes the conditional formulas $E[S \mid N = n] = n\mu$ and $\operatorname{Var}(S \mid N = n) = n\sigma^2$ of the next cards hold at $n = 0$ as well.

Definition Let $X_1, X_2, \dots$ be identically distributed with finite mean $\mu$, and let $N$ be a random count with $E[N] < \infty$ that is independent of the whole sequence. Then

For each $n$ with $P(N = n) > 0$, independence and linearity give $E[S \mid N = n] = E[X_1] + \dots + E[X_n] = n\mu$, and the empty sum gives the same at $n = 0$. So $E[S \mid N] = N\mu$, and the tower rule gives $E[S] = E[N\mu] = E[N]\,\mu$. The same argument applied to the nonnegative terms $\lvert X_i \rvert$ shows that $E[\lvert S \rvert] \le E[N]\, E[\lvert X_1 \rvert] < \infty$, so $S$ has a finite mean and the tower rule applies. This formula does not need the terms to be independent of each other, and the terms may take negative values.

Definition Let $X_1, X_2, \dots$ be IID with mean $\mu$ and finite variance $\sigma^2$, and let $N$ be a random count with $E[N^2] < \infty$ that is independent of the whole sequence. Then

For each $n$ with $P(N = n) > 0$, the terms $X_1, \dots, X_n$ are still IID given $N = n$, so their variances add and $\operatorname{Var}(S \mid N = n) = n\sigma^2$. Together with $E[S \mid N = n] = n\mu$ this gives $E[S^2] = E[N]\,\sigma^2 + E[N^2]\,\mu^2 < \infty$. The law of total variance with $N$ as the group label then gives the within-group variation $E[N\sigma^2] = E[N]\,\sigma^2$ and the between-group variation $\operatorname{Var}(N\mu) = \mu^2 \operatorname{Var}(N)$. The step $\operatorname{Var}(S \mid N = n) = n\sigma^2$ uses the independence of the terms from each other as well as from $N$.