week 5
1
Theorem
For
g
:
R
Æ
R
•
If
X
is a discrete random variable then
•
If
X
is a continuous random variable
•
Proof:
We proof it for the discrete case. Let
Y
=
g
(
X
) then
(
)
[
]
( )
( )
∑
=
x
X
x
p
x
g
X
g
E
(
)
[
]
( )
( )
∫
∞
∞
−
=
dx
x
f
x
g
X
g
E
X

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week 5
2
Example to illustrate steps in proof
•
Suppose
i.e.
and the possible values of
X
are
so the possible values of
Y
are
then,
3
,
2
,
1
:
±
±
±
X
2
X
Y
=
( )
2
x
x
g
=
9
,
4
,
1
:
Y