Lecture 5 on BST 631: Statistical Theory I – Kui Zhang, 09/02/2008
1
Review for the previous lecture
Definition:
cdf, pmf, pdf, identically distributed
Theorem:
How to determine the cdf, pmf, or pdf.
Example
: how to calculate cdf, pmf, or pdf.
Chapter 2 – Transformations and Expectations
Chapter 2.1 – Distributions of Functions of a Random Variable
Problem:
Let
X
be a random variable with cdf
()
X
Fx
. If we define any function of
X
, say
Yg
X
=
,
then
X
=
is also a random variable whose distribution depends on
X
F
and the function
g
. Specifically, for any set
A
,
(
(
)
)
.
PY A
Pg X
A
∈
=∈
Formally, if
X
∈
X
and
Y
∈
Y
, then
(
)
gX
is a mapping from the sample space of
X
,
X
, to the sample space of
Y
,
Y
, i.e.,
( ):
gx
→
XY
.
To go from
Y
back to
X
, we define the inverse function of
g
, denoted by
1
g
−
, as
1
() {
:() }
gA
x
g
x
A
−
=
∈∈
X
.
If
A
is a set that only contains a single point, say
{ }
A
y
=
, then
1
: ()
}
gy
x
g
x
y
−
=
∈=
X
.