Question

Find mean and standard deviation of the continuous random variable $X$ whose p.d.f is given by $f\left(x\right)=6x(1-x)$ for 0<x<1 and 0 otherwise.

Answer 1

$f\left(x\right)=6x(1-x)$

Mean $E\left(X\right)={\int }_0^{1}xf\left(x\right)dx=6{\int }_0^1{x}^2(1-x)dx=6[\frac{x^3}{3}-\frac{x^4}{4}{]}_0^1=6\times \frac{1}{12}=\frac{1}{2}$

$E\left(X^2\right)={\int }_0^1{x}^{2}f\left(x\right)dx={\int }_0^{1}6x^3(1-x)dx=6[\frac{x^4}{4}-\frac{x^5}{5}]=6\times \frac{1}{20}=\frac{3}{10}$

Variance ${\sigma }^2=E\left(X^2\right)-\left(E\right(X){)}^2=\frac{3}{10}-\frac{1}{4}=\frac{1}{20}$

Standard deviation is $\sigma =\frac{1}{\sqrt{20}}$