Largest variance for a variable bounded in minus one to one

Question

A random variable $Y$ takes values only in the interval $[-1,1]$ . What is the greatest value $\operatorname{Var}(Y)$ can have?

Short answer

$1$ . Always $\operatorname{Var}(Y)\le\mathbb{E}[Y^2]\le1$ since $|Y|\le1$ . Equality is attained by $Y=\pm1$ with probability $\tfrac12$ each, which has mean $0$ and $Y^2=1$ .

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
Covariance and Correlation

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