Question

The Probability distribution of $X$, the number of defects per $10$ meters of a fabric is given by

$x$	$0$	$1$	$2$	$3$	$4$
$P(X=x)$	$0.45$	$0.35$	$0.15$	$0.03$	$0.02$

Find the variance of $X$.

Answer 1

$xi$	$P(X=x)$	$xiPi$	$x{i}^2$	$x{i}^{2}Pi$
0	0.45	0	0	0
1	0.35	0.35	1	0.35
2	0.15	0.30	4	0.6
3	0.03	0.09	9	0.27
4	0.02	0.08	16	0.32
		0.82		1.54

$E\left(x\right)={\sum }^{}{x}_i{P}_i=0.82$

$E\left(x^2\right)={\sum }^{}{x}_i^2{P}_i=1.54$

Therefore,

$Variance=E\left(x^2\right)-\left[E\right(x){]}^2$

$Variance=1.54-[0.82{]}^2$

$Variance=1.54-0.6724$

$Variance=0.8676$

Hence, this is the answer.