368 CHAPTER 6 Inferential Statistics

∫t−a

−∞ fX(x)dx =

∫t

−∞fX(x−a)dx.

In other words, for all real numberst, we have

∫t

−∞fY(x)dx =

∫t

−∞fX(x−a)dx.

This implies (e.g., by the Fundamental Theorem of Calculus) that

fX+a(x) = fX(x−a) (∗)

for allx∈R.

Next, we would like to compute the density function for the random
variableY =X^2 in terms of that ofX. To do this, note that

∫t

0 fX

(^2) (x)dx = P(X^2 < t) = P(−

√

t < X <

√

t) =

∫√t
−√tf(x)dx.
An application of the Fundamental Theorem of Calculus gives

fX^2 (x) =

1

2

√

x

(

fX(

√

x)−fX(−

√

x)

)

. (∗∗)

Using equations (*) and (**), we can compute the variance of the con-
tinuous random variableXhaving meanμ, as follows. We have

Var(X) = E((X−μ)^2 )

=

∫∞

0 xf(X−μ)^2 (x)dx

by (**)=^1

2

∫∞

0

√

xfX−μ(

√

x)dx −

1

2

∫∞

0

√

xfX−μ(−

√

x)dx

(u=√x)

=

∫∞

0 u

(^2) fX−μ(u)du+
∫ 0
−∞u
(^2) fX−μ(u)du=
∫∞
−∞u
(^2) fX−μ(u)du
by (*)= ∫∞
−∞
u^2 fX(u+μ)du

∫∞
−∞(u−μ)
(^2) fX(u)du,

∫∞
−∞(x−μ)
(^2) fX(x)dx.