Question

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

(i)

X = x	1	2	3
P(X = x)	1/5	2/5	2/5

(ii)

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

(iii)

X = x	1	2	3	...	n
P(X = x)	1/n	1/n	1/n	...	1/n

(iv)

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

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

From the table,

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

xi	P(xi)	xi.P(xi)	xi2.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,

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

From the table,

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

xi	P(xi)	xi.P(xi)	xi2.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,