2
$\begingroup$

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 $z=\sum_1^m y_i$ ?

$\endgroup$
2
$\begingroup$

We have $Var\,z=m\,Var\,y_1$, $$Var\,y_1=P(x_1\ge K)(1-P(x_1\ge K))=P(x_1\ge K)P(x_1<K))=(1-F(K-1))F(K-1) =(1-I_{1-p}(n-K+1,K))I_{1-p}(n-K+1,K)$$ 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, $$Var\,z=m\,(1-I_{1-p}(n-K+1,K))I_{1-p}(n-K+1,K).$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.