Maximum variance on an interval

Let $Y$ be any random variable taking values only in $[-1, 1]$ . What is the largest possible value of $\operatorname{Var}(Y)$ ?

Show hints (2)+

$\operatorname{Var}(Y)\le\mathbb{E}[(Y-0)^2]=\mathbb{E}[Y^2]\le 1$ since $|Y|\le1$ .
Equality: $Y=\pm1$ each with probability $1/2$ (mean $0$ , $\mathbb{E}[Y^2]=1$ ).

Answer

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