Variance of Poisson distributed variables $\leq1$

Question

If $Z_n$ is the number of sample observations of $X\leq1$ where $X \sim \text{Poisson}(\mu)$, then what is $\text{Var}(\frac{Z_n}{n}$)?

Answer 1

Hints:

• $\text{Var}(Z_n/n) = \frac{1}{n^2} \text{Var}(Z_n)$
• $Z_n$ can be written as $I_1 + I_2 + \cdots + I_n$ where $I_j$ is a random variable that equals $1$ if $X_j \le 1$, and equals $0$ otherwise.
• If $I_1, \ldots, I_n$ are independent then $\text{Var}(I_1 + \cdots + I_n) = \text{Var}(I_1) + \cdots + \text{Var}(I_n)$.