Question

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

x = x	-1	0	3
P (X = x)	`1/5`	`2/5`	`2/5`

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)
-1	`1/5`	`-1/5`	`1/5`
0	`2/5`	0	0
1	`2/5`	`2/5`	`2/5`
Total	1	`1/5`	`3/5`

From the table,

∑x_i·P (x_i) = `1/5`,

∑ x_i ²·P (x_i) = `3/5`

Expected value = E (X) = ∑x_i·P (x_i) = `1/5` = 0.2

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

= `3/5 - (1/5)^2`

= `3/5 - 1/25`

= `15/25 - 1/25`

= `14/25`

= 0.56

Standard deviation = `sqrt(V(X)`

= `sqrt(0.56)`

= 0.7483