Introduction to Probability and Statistics for Engineers and Scientists

4.6Variance 119

Because we expectXto take on values around its meanE[X], it would appear that
a reasonable way of measuring the possible variation ofXwould be to look at how far
apartXwould be from its mean on the average. One possible way to measure this would
be to consider the quantityE[|X−μ|], whereμ=E[X], and[X−μ] represents the
absolute value ofX−μ. However, it turns out to be mathematically inconvenient to deal
with this quantity and so a more tractable quantity is usually considered — namely, the
expectation of the square of the difference betweenXand its mean. We thus have the
following definition.

Definition

IfXis a random variable with meanμ, then thevarianceofX, denoted by Var(X), is
defined by

Var(X)=E[(X−μ)^2 ]

An alternative formula for Var(X) can be derived as follows:

Var(X)=E[(X−μ)^2 ]

=E[X^2 − 2 μX+μ^2 ]

=E[X^2 ]−E[ 2 μX]+E[μ^2 ]

=E[X^2 ]− 2 μE[X]+μ^2

=E[X^2 ]−μ^2

That is,

Var(X)=E[X^2 ]−(E[X])^2 (4.6.1)

or, in words, the variance ofXis equal to the expected value of the square ofXminus the
square of the expected value ofX. This is, in practice, often the easiest way to compute
Var(X).

EXAMPLE 4.6a Compute Var(X) whenXrepresents the outcome when we roll a fair die.

SOLUTION SinceP{X=i}=^16 ,i=1, 2, 3, 4, 5, 6, we obtain

E[X^2 ]=

∑^6

i− 1

i^2 P{X=i}

= 12

( 1

6

)

+ 22

( 1

6

)

+ 32

( 1

6

)

+ 42

( 1

6

)

+ 52

( 1

6

)

+ 62

( 1

6

)

=^916