Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

1.9. Some Special Expectations 69

It is customary to callσ(the positive square root of the variance) thestandard
deviationofX (or the standard deviation of the distribution). The numberσ
is sometimes interpreted as a measure of the dispersion of the points of the space
relative to the mean valueμ. If the space contains only one pointkfor which
p(k)>0, thenp(k)=1,μ=k,andσ=0.
While the variance is not a linear operator, it does satisfy the following result:

Theorem 1.9.1. LetXbe a random ravariable with finite meanμand variance
σ^2. Then for all constantsaandb,

Var(aX+b)=a^2 Var(X). (1.9.1)

Proof.BecauseEis linear,E(aX+b)=aμ+b. Hence, by definition

Var(aX+b)=E

{

[(aX+b)−(aμ+b)]^2

}

=E

{

a^2 [X−μ]^2

}

=a^2 Var(X).

Based on this theorem, for standard deviations,σaX+b=|a|σX. The following
example illustrates these points.

Example 1.9.1.Suppose the random variableXhas a uniform distribution, (1.7.4),
with pdffX(x)=1/(2a),−a<x<a, zero elsewhere. Then the mean and variance
ofXare:

μ =

∫a

−a

x

1

2 a

dx=

1

2 a

x^2

2

∣

∣∣

∣

a

−a

=0,

σ^2 =

∫a

−a

x^2 =

1

2 a

x^3

3

∣

∣

∣

∣

a

−a

=

a^2

3

.

so thatσX=a/

√
3 is the standard deviation of the distribution ofX.Consider
the transformationY =2X. Because the inverse transformation isx=y/2and
dx/dy=1/2, it follows from Theorem 1.7.1 that the pdf ofY isfY(y)=1/ 4 a,
− 2 a<y< 2 a, zero elsewhere. Based on the above discussion,σY =(2a)/

√
3.
Hence, the standard deviation ofYis twice that ofX, reflecting the fact that the
probability forY is spread out twice as much (relative to the mean zero) as the
probability forX.

Example 1.9.2.LetXhave the pdf

f(x)=

{ 1

2 (x+1) −^1 <x<^1

0elsewhere.

Then the mean value ofXis

μ=

∫∞

−∞

xf(x)dx=

∫ 1

− 1

x

x+1

2

dx=

1

3

,

while the variance ofXis

σ^2 =

∫∞

−∞

x^2 f(x)dx−μ^2 =

∫ 1

− 1

x^2

x+1

2

dx−

(

1

3

) 2

=

2

9

.