# Sum of indicator functions of binomial random variables

Let $$x_1, x_2,..., x_m$$ be iid binomial random variables (each with a number of trials n and probability of success in each trial p). Define a list of binary indicator variables $$y_1,y_2,...,y_m$$ for at least $$K$$ successes i.e.,

$$y_i = \begin{cases} 1 & \text{if } x_i\geq K\\ 0 & \text{else} \end{cases}$$

What is the variance of $$z=\sum_1^m y_i$$ ?

## 1 Answer

We have $$Var\,z=m\,Var\,y_1$$, $$Var\,y_1=P(x_1\ge K)(1-P(x_1\ge K))=P(x_1\ge K)P(x_1 according to this, where $$F$$ is the cdf of the binomial distribution with parameters $$n$$ and $$p$$, and $$I_\cdot(\cdot,\cdot)$$ is the regularized incomplete beta function. Thus, $$Var\,z=m\,(1-I_{1-p}(n-K+1,K))I_{1-p}(n-K+1,K).$$