# 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) 132 532 1032 1032 532 132 - Mathematics and Statistics

Sum

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

#### Solution

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,

∑ xi . P (xi)  = 80/32 = 5/2 ,
∑ xi2 . P ( xi) = 240/32 = 15/2

Expected value = E (X) = ∑ xi . P (xi

= 5/2 = 2.5

Variance = V(x) = ∑ xi2 . P (xi) - [ ∑ xi·P (xi) ]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

Concept: Variance of a Random Variable
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#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 7 Probability Distributions
Miscellaneous Exercise 2 | Q 10.4 | Page 244