91Ó°ÊÓ

\({\rm{X}} \sim {\rm{Uniform}}\left( {0,1} \right)\).

The PDF of the random variable X is:

\(f\left( x \right) = \left\{ {\begin{aligned}{*{20}{c}}{1;{\rm{ if }}0 < x < 1}\\{0;\,{\rm{otherwise}}}\end{aligned}} \right.\)

\(\begin{aligned}{}E\left( x \right) &= \int\limits_0^1 {xdx} \\ &= \left( {\frac{{{x^2}}}{2}} \right)_0^1\\ &= \frac{1}{2}\end{aligned}\)

Therefore, \(E\left( X \right) = \frac{1}{2}\).

\(\begin{aligned}{}E\left( {{X^2}} \right) &= \int\limits_0^1 {{x^2}dx} \\ &= \left( {\frac{{{x^3}}}{3}} \right)_0^1\\ &= \frac{1}{3}\end{aligned}\)

Therefore, \(E\left( {{X^2}} \right) = \frac{1}{3}\).

We know,

\(\begin{aligned}{}Var\left( X \right) &= E\left( {{X^2}} \right) - {\left( {E\left( X \right)} \right)^2}\\ &= \frac{1}{3} - {\left( {\frac{1}{2}} \right)^2}\\ &= \frac{1}{3} - \frac{1}{4}\\ &= \frac{1}{{12}}\end{aligned}\)

Thus, \(Var\left( X \right) = \frac{1}{{12}}\).