Question

find the variance and standard deviation of the probability distribution of the random variable W if P{W = w} = {w+1}/20 for W = {1, 2, 3, 4, 5}

Answer 1

$V(W)=M(W^2)-M^2(W)$

$P(W=1)={\frac 2 {20}}$

$P(W=2)={\frac 3 {20}}$

$P(W=3)={\frac 4 {20}}$

$P(W=4)={\frac 5 {20}}$

$P(W=5)={\frac 6 {20}}$

$M(W)={\frac 2 {20}}*1+{\frac 3 {20}}*2+{\frac 4 {20}}*3+{\frac 5 {20}}*4+{\frac 6 {20}}*5={\frac {70}{20}}=3.5\implies M^2(W)=12.25$

$M(W^2)={\frac 2 {20}}*1^2+{\frac 3 {20}}*2^2+{\frac 4 {20}}*3^2+{\frac 5 {20}}*4^2+{\frac 6 {20}}*5^2={\frac {280}{20}}=14$

So, $V(W)=14-12.25=1.75$

$\sigma(W)=\sqrt{V(W)}=\sqrt{1.75}\approx1.323$