SECTION NOTE 5
ORIE360/560, FALL 2004
CHONG WANG
1.
Expected Values
Expectation is one of the most important concepts in probability theory.
Deﬁnition:
For a discrete random variable
X
with a pmf
p
X
, the expected value is deﬁned by
E
[
X
] =
X
x
xp
X
(
x
)
.
For a continuous radom variable
X
with a pdf
f
X
, the expected value is deﬁned by
E
[
X
] =
Z
∞
∞
xf
X
(
x
)
dx.
Example
:(Problem 25, Chapter 4)
A total of 4 buses carrying 148 students from the same school arrive at a football
stadium. The buses carry, respectively, 40,33,25, and 50 students. One of the students
is randomly selected. Let
X
denote the number of students that were on the bus
carrying this randomly selected student. One of the 4 bus drivers is also randomly
selected. Let
Y
denote the number of students on her bus.
(a) Which of
EX
or
EY
do you think is larger? Why?
(b) Compute
EX
and
EY
.
Solution
:
(a)
EX
is larger. The reason is for
Y
, no matter how many passengers on the bus,
the four buses will be chosen with the same chances. But for
X
, the more passengers
on the bus, the bigger chance the bus will get chosen. So relatively large values of
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 Summer '08
 WEBER
 Probability theory, ex, CHONG WANG

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