Stat 311 Introduction to Mathematical
Statistics
Zhengjun Zhang
Department of Statistics, University of Wisconsin, Madison, WI 53706, USA
November 35, 2009
Functions of a Random Variable
Theorem 5.1
Let
X
be a continuous random variable, and suppose that
φ
(
x
) is a strictly
increasing function on the range of
X
.
Define
Y
=
φ
(
X
).
Suppose that
X
and
Y
have
cumulative distribution functions
F
X
and
F
Y
respectively. Then these functions are related
by
F
Y
(
y
) =
F
X
(
φ

1
(
y
))
.
If
φ
(
x
) is strictly decreasing on the range of
X
, then
F
Y
(
y
) = 1

F
X
(
φ

1
(
y
))
.
Corollary 5.1
Let
X
be a continuous random variable, and suppose that
φ
(
x
) is a strictly
increasing function on the range of
X
. Define
Y
=
φ
(
X
). Suppose that the density functions
of
X
and
Y
are
f
X
and
f
Y
, respectively. Then these functions are related by
f
Y
(
y
) =
f
X
(
φ

1
(
y
))
d
dy
φ

1
(
y
)
.
If
φ
(
x
) is strictly decreasing on the range of
X
, then
f
Y
(
y
) =

f
X
(
φ

1
(
y
))
d
dy
φ

1
(
y
)
.
Corollary 5.2
If
F
(
y
) is a given cumulative distribution function that is strictly increasing
when 0
< F
(
y
)
<
1 and if
U
is a random variable with uniform distribution on [0
,
1], then
Y
=
F

1
(
U
)
has the cumulative distribution
F
(
y
).
Simulation
0
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
•
This corollary tells that to simulate a random variable with a given cumulative distri
bution
F
we need to simulate a uniform random variable first, then convert the value
into the desired scale using the inverse distribution function.
Example A number
U
is chosen at random in the interval [0
,
1]. Find the distribution functions
and density functions for the following transformed random variables.
1.
R
=
U
2
.
2.
S
=
U
(1

U
).
3.
T
=
U/
(1

U
).
Normal density: The bell shape curve
4
2
2
4
0.1
0.2
0.3
0.4
σ=1
σ=2
Figure 1:
Normal density function
f
X
(
x
) =
1
√
2
πσ
e

(
x

μ
)
2
/
2
σ
2
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Staff
 Statistics, Normal Distribution, Probability theory, probability density function, FY

Click to edit the document details