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Lecture 23.
Some topics from 10.1
Fact.
Expectation is linear: E(
aX
) =
a
E(
X
) for any constant
a
and any random variable
X
and E(
X
+
Y
) = E(
X
) + E(
Y
) for any random variables
X
and
Y
. Thus, for any
constants
a
i
and random variables
X
i
,
E(
a
1
X
1
+
. . .
+
a
n
X
n
) =
a
1
E(
X
1
) +
. . .
+
a
n
E(
X
n
)
.
Comment.
Of course, for the statements in the Fact to be a meaningful, it is necessary
that E(
X
), E(
Y
) and all the E(
X
i
) exist. In the following, we will deal only with random
variables that have and expected value.)
Question.
Give an example of a discrete random variable that has no expected value.
Example.
If a die is rolled
n
times, the expected sum of the rolls is
n
time the expected
outcome of a single roll. More generally, if any experiment with numerical outcomes is
repeated
n
times, the expected sum of the outcomes is
n
times the expected outcome of a
single trial.
Example 10.3.
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 Spring '08
 Britt
 Probability

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