Linear Combinations of Poisson Random Variables

When two independent Poisson random variables are added, the result is again Poisson, with rate equal to the sum of the individual rates.

Sum of Independent Poissons. If $X_1 \sim \text{Poisson}(\lambda_1)$ and $X_2 \sim \text{Poisson}(\lambda_2)$ are independent, then

$$X_1 + X_2 \sim \text{Poisson}(\lambda_1 + \lambda_2).$$

Intuitively, if events of type 1 occur randomly at rate $\lambda_1$ and events of type 2 occur independently at rate $\lambda_2,$ then events of either type occur at the combined rate $\lambda_1 + \lambda_2.$

For example, suppose emails arrive in your inbox from coworkers at rate $\lambda_1 = 4$ per hour and from customers at rate $\lambda_2 = 6$ per hour, independently. Then the total number of emails $T = X_1 + X_2$ in one hour satisfies

$$T \sim \text{Poisson}(10),$$

so

$$P(T = 0) = \dfrac{e^{-10} \cdot 10^0}{0!} = e^{-10}.$$