Question

Find the variance and standard deviation of the random variable $X$ whose probability distribution is given below:

$x$	$0$	$1$	$2$	$3$
$P(X=x)$	$\frac{1}{8}$	$\frac{3}{8}$	$\frac{3}{8}$	$\frac{1}{8}$

Answer 1

Let mean of $X$ be $\mu$

$\therefore \mu ={\sum }_{x=0}^{3}X\times P\left(X\right)=\frac{12}{8}=\frac{3}{2}$

$\therefore \text{Var}\left(X\right)={\sigma }^2={\sum }_{x=0}^3{X}^2\times P\left(X\right)-{\mu }^2=\frac{24}{8}-\frac{9}{4}=\frac{3}{4}$

$\therefore \text{Standard Deviation}=\sigma =\sqrt{\text{Var}\left(X\right)}=\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2}$

These are the required answers.