UCLA
EE131A (KY)
1
EE 131A
Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 101
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UCLA
EE131A (KY)
2
Expectation of a function g(X) of X (1)
•
So far, we have considered the expectation E{.} of
g(X) = X to yields the mean
;
g(X) = X
2
to yield
the second moment; and g(X) = (X
)
2
to yield the
variance
2
.
•
The
nth moment
m
n
is the expectation of X
n
defined by
•
We note, using this new notation, the mean
= m
1
is
the first moment.
n
n
n

m
= E
X
=
x
f(x) dx .
UCLA
EE131A (KY)
3
Expectation of a function g(X) of X (2)
•
Consider a rv
X with a pdf
f
X
(x) and a function
g(x).
Define a new rv
Y = g(X).
The expectation
of E{Y}=
Y
can be obtained two ways.
•
Method 1 –
Since Y is a rv, it has a pdf
f
Y
(y) and a
cdf
F
Y
(y) (considered earlier in Lec82). Then
•
Method 2 –
Treat Y = g(X) as a function of X.
Y
Y

μ
E{Y} =
y f
(y) dy .
(1)
Y
X

μ
E{g(X)} =
g(x) f
(x) dx .
(2)
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