Question

find the variance and standard deviation of the probability distribution of a random variable x which can take only the values 2,4,5 and 9, given that p(2)= 9/20, p(4)= 1/20, p(5) = 1/5, and p(3)= 3/10

Answer 1

Given the probability distribution,

x 2 4 5  9

p(x) 9/20 1/20 1/5 3/10

The expected value is

$E(x)=\sum xp(x)=(2\times{9\over20})+(4\times{1\over20})+(5\times{1\over5})+(9\times{3\over10})=4.8$

The variance is given as,

$var(x)=\sum (x^2)-(\sum(x))^2$  

We need to find $E(x^2)=\sum x^2p(x)=)=(4\times{9\over20})+(16\times{1\over20})+(25\times{1\over5})+(81\times{3\over10})=31.9$

Now,

$var(x)=31.9-(4.8)^2=31.9-23.04=8.86$

The standard deviation is,

s$sd(x)=\sqrt{var(x)}=\sqrt{8.86}=2.9766$