Linear Combinations of Binomial Random Variables

When two binomial random variables share the same success probability $p$ and are independent, their sum is also binomial.

Theorem. If $X \sim \text{Bin}(n, p)$ and $Y \sim \text{Bin}(m, p)$ are independent, then

$$X + Y \sim \text{Bin}(n + m,\, p).$$

Intuitively, $X$ counts successes in $n$ independent trials with success probability $p$, and $Y$ counts successes in $m$ more independent trials with the same $p$. Pooling all $n + m$ trials gives a binomial with parameter $n + m$ and the same $p$.

For example, if $X \sim \text{Bin}(3, 0.4)$ and $Y \sim \text{Bin}(5, 0.4)$ are independent, then

$$X + Y \sim \text{Bin}(8,\, 0.4).$$

Two warnings.

1. The success probabilities must match. If $X \sim \text{Bin}(3, 0.4)$ and $Y \sim \text{Bin}(5, 0.7)$, then $X + Y$ is not binomial.
2. The result requires independence.