Question

Find the expected value, variance, and standard deviation of the random variable whose p.m.f.’s are given below :

x = x	0	1	2	3	4	5
P (X = x)	`1/32`	`5/32`	`10/32`	`10/32`	`5/32`	`1/32`

Answer 1

We construct the following table to find the expected value, variance and standard deviation:

xi	P (xi)	xi·P (xi)	xi 2·P (xi) = xi × xi·P (xi)
0	`1/32`	0	0
1	`5/32`	`5/32`	`5/32`
2	`10/32`	`20/32`	`40/32`
3	`10/32`	`30/32`	`90/32`
4	`5/32`	`20/32`	`80/32`
5	`1/32`	`5/32`	`25/32`
Total	`32/32`	`80/32`	`240/32`

From the table,

∑ x_i . P (x_i)  = `80/32 = 5/2` ,
∑ x_i² . P ( x_i) = `240/32 = 15/2`

Expected value = E (X) = ∑ x_i . P (x_i) 

= `5/2` = 2.5

Variance = V(x) = ∑ x_i² . P (x_i) - [ ∑ x_i·P (x_i) ]²

= `15/2 - (5/2)^2`

= `15/2 - 25/4`

= `30/4 - 25/4`

= `5/4`

= 1.25

Standard deviation = `sqrt(V(X)) = sqrt 1.25`
= 1.118