The random variable $X$ has a binomial distribution $b(n,p)$. For what value of $p$ is the standard deviation of $X$ the greatest (note: the answer is independent of $n$).
Can someone help with this question! I know the formula of standard deviation, it's $\sqrt{n p( 1 - p)}$. My question is: Are we trying to find a value closest to zero? Can anyone explain and help to find the value of $p$?
Hint: What you want to do is find $p\in[0,1]$ such that the function $$ f\colon[0,1]\to\mathbb R, \quad p \mapsto np(1-p),$$ attains its maximum in $p$. You can use straight-forward calculations or brute force (this function is twice differentiable).