Some facts about expectation
October 22, 2010
1
Expectation identities
There are certain useful identities concerning the expectation operator that
I neglected to mention early on in the course. Now is as good a time as any
to talk about them. Here they are:
•
Suppose that
X
is a continuous random variable having pdf
f
(
x
), and
suppose that
α
(
x
) is some function. Then,
E
(
α
(
X
)) =
integraldisplay
∞
∞
α
(
x
)
f
(
x
)
dx.
There is also a discrete analogue of this identity.
•
Suppose that
X
1
, ..., X
k
are random variables and that
λ
1
, ..., λ
k
are
real numbers. Then,
E
(
λ
1
X
1
+
· · ·
+
λ
k
X
k
) =
λ
1
E
(
X
1
) +
· · ·
+
λ
k
E
(
X
k
)
.
This property is called “Linearity of Expectation”.
•
Suppose that
X
1
, ..., X
k
are independent random variables. Then, the
expectation of a product is the product of expectations; that is
E
(
X
1
· · ·
X
k
) =
E
(
X
1
)
E
(
X
2
)
· · ·
E
(
X
k
)
.
Note:
You really do need independence here. There are examples of
r.v.’s that are not independent, for which this identity fails to hold.
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 Spring '08
 Staff
 Statistics, Probability, Probability theory, 1 k, yG, X1, xk

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