Which random variable $X:\Omega\rightarrow \mathbb R$ has the maximal standard deviation for the two following cases:
- $\Omega$ is descrete and, let's $n\in \mathbb N$ such that $|\Omega|=n$
- $\Omega$ is $[0;1]$
I believe standard deviation $\sigma$ of random variable $X$ is $\sigma(X) = \sqrt{\mathbb E[X²]-\mathbb E[X]^2}$. So our problem would be to find
$$\displaystyle\arg \max_{X\in \mathbb R^\Omega} \mathbb E[X^2] - \mathbb E[X]^2$$
How should I continue from there?