Question

A random variable is uniformly distributed over the interval 2 to 10. Its variance will be

• A. 16/3
• B. 6
• C. 256/9
• D. 36

Answer 1

The correct option is A 16/3

$\text{Given}xϵ[2,10]$ so p.d.f. for uniform random variable is

$f\left(x\right)=(\begin{array}{cc}\frac{1}{10-2},& 2<x<10\\ 0,& \mathrm{otherwise}\end{array}$
we know that
$Var\left(x\right)=E\left(x^2\right)-{\left[E\right(x\left)\right]}^2$
$E\left(x\right)={\int }_{-\infty }^{\infty }xf\left(x\right)dx={\int }_2^{10}x\left(\frac{1}{8}\right)dx=6$

& $E\left(x^2\right)={\int }_{-\infty }^{\infty }{x}^{2}f\left(x\right)dx={\int }_2^{10}{x}^2\left(\frac{1}{8}\right)dx$

$=\frac{124}{3}$

So $Var\left(x\right)=E\left(x^2\right)-{\left[E\right(x\left)\right]}^2=\frac{124}{3}-(6{)}^2$

$=\frac{16}{3}$