Variance of a standard uniform variable

Question

$X$ has density $1$ on $[0,1]$ and $0$ elsewhere. What is $\operatorname{Var}(X)$ ?

Short answer

$1/12$ . $\mathbb{E}X=\tfrac12$ and $\mathbb{E}X^2=\int_0^1x^2dx=\tfrac13$ , so the variance is $\tfrac13-\tfrac14=\tfrac1{12}$ .

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Lessons behind this question

Covariance and Correlation
Continuous Random Variables and Probability Density Functions
Cumulative Distribution Functions

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