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Homework Set #6
Problem 1
In planning a building, the number of elevators is chosen on the basis of balancing initial
costs versus the expected delay times of the users. These delays are closely related to the
number of stops the elevator makes on a trip. If an elevator runs full (n people) and there
are k floors, we want to find the expected number of stops R the elevator makes on any
trip. Assuming that the passengers act independently and that any passenger chooses a
floor with equal probability
k
1
. Show that:
⎡
−
⎛
1
−
⎜
⎝
1
k
⎞
⎟
⎠
n
⎤
⎥
⎥
⎦
E[R]
=
1
k
⎢
⎢
⎣
Hint:
It is often useful to define “indicator random variables” as follows: Let X
i
= 1 if
k
the elevator stops at floor i, and X
i
= 0 if it does not. Then observe that
R
∑
=
X
i
. Find
=
1
the expected value of X
i
after finding the probability that X
i
= 0.
Problem 2
In the very preliminary planning of some harbor island developments, there was
discussion regarding the cost estimates of building four bridges. There had as yet not
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This note was uploaded on 12/04/2011 for the course ESD 1.151 taught by Professor Danieleveneziano during the Spring '05 term at MIT.
 Spring '05
 DanieleVeneziano

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