distribution of the indicator function of poisson

Question

Let $X_1,\dots,X_n$ be independent random variables with poisson distribution

Given indicator function $$ U_i=\left\{ \begin{aligned} 1 && X_1 \ = 0\\ 0 && X_1 >0 \end{aligned} \right. $$

What is the distribution of $\sum\limits_{i=1}^n U_i?$

Answer 1

The distribution of $\sum_{i=1}^nU_i$ is the Binomial distribution with parameters $n$ and $p=\Pr\{X_1=0\}=\frac{\lambda^0e^{-\lambda}}{0!}=e^{-\lambda}$ since $U_1,\ldots,U_n$ are independent and identically distributed Bernoulli random variables.