\(f(x) = \frac x{a^2 }e^{\frac{-x^2}{2a^2}}, 0\le x < \infty\)
\(\therefore Var(X) = E(X^2) - (E(X))^2\)
\(= \int\limits ^\infty_0 x^2 f(x)dx - \left[\int\limits^\infty_0 xf(x)dx\right]^2\)
\(= \int\limits^\infty_0\frac{x^3}{a^2}\,e^{-\frac{x^2}{2a^2}}dx-\left[\int\limits^\infty_0\frac{x^2}{a^2}\,e^{-\frac{x^2}{2a^2}}dx \right]^2\)
Let \(\frac{x^2}{2a^2} = t\)
⇒ \(x^2 = 2a^2 t\)
⇒ \(2x\,dx = 2a^2dt\)
⇒ \(x\,dx = a^2 dt\)
\(= \int\limits ^\infty_0\frac{a^2}{a^2}\times 2a^2 t \,e^{-t}dt - \left[\int\limits^\infty_0\frac{\sqrt2 a\sqrt t}{a^2}\times a^2 e^{-t}dt\right]^2\)
\(= 2a^2 \int\limits^\infty_0 t^{2-1}e^{-t}dt - \left[\sqrt2 a\int\limits^\infty_0 t^{\frac32 - 1}\, e^{-t}dt\right]^2\)
\(= 2a^2 \Gamma(2)- \left(\sqrt2 a \Gamma\left(\frac32\right)\right)^2\) \(\left(\because \int\limits ^\infty_0 x^{n- 1}e^{-x}dx = \Gamma(n)\right)\)
\(= 2a^2 - \left(\sqrt2a.\frac12\Gamma\left(\frac12\right)\right)^2\) \(\begin{pmatrix}\because \Gamma \left(\frac32\right)= \Gamma\left(\frac12 + 1\right)= \frac12\Gamma\left(\frac12 \right)\\\because\Gamma(2) = 1! = 1\end{pmatrix}\)
\(= 2a^2 - \left(\frac{a}{\sqrt2}\sqrt{\pi}\right)^2\) \((\because \Gamma \left(\frac12 \right) = \sqrt{\pi})\)
\(= 2a^2 - \frac{\pi a^2}{2} \)
\(= a^2 \left(2 - \frac{\pi}2\right)\)
\(= \left(\frac{4 - \pi}{2}\right)a^2\)
Var(X) = \(\left(\frac{4 - \pi}2\right)a^2\)
