1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J
Logistical and Transportation Planning Methods
Problem Set #1
Issued: September 11, 2006
Due: September 20, 2006
Problem 1
Twohorse race
Consider a longdistance, twohorse race between horses A and B. The finishing
time of Horse A is denoted by random variable (r.v.) X, which is uniformly distributed
between 3 and 8 minutes. Horse B’s finishing time, Y, is uniformly distributed in [x/4,
2x] given that X=x. In other words, the conditional pdf for Y given X=x is:
4
x
f
Y

X
(
y

x
) =
for
≤
y
≤
2
x
.
7
x
4
(a) Please give the general relationship between the compound pdf of two arbitrary
r.v.’s
U
and
V
, the conditional pdf of
U
given
V
=
v
and the pdf of
V
.
Then give the general relationship between the compound pdf of
U
and
V
and the
marginal probability density function of Y.
(b) Derive the pdf of Y. (Hint: there are 3 possible cases to consider.)
(c) Find
f
X

Y
(
x

y
)
, the conditional pdf of X given Y.
(d) Let S be the event “Horse A wins the race”. Is±
P
(
S

Y
=
y
)
=
P
(
S
)
, i.e. are the
event S and r.v. Y independent?
(e) What is the probability that the winner will win by less than 1 minute?
(f) What is the probability that the winner’s time will be less than 6 minutes?
Problem 2
Cell Phones
You are buying a cell phone and must select a monthly calling plan.
The possible monthly calling plans are numbered from 1 to N (N is finite!). Plan
i
costs
D
i
dollars per month and offers
M
i
free minutes of cell phone conversation per
month with
C
i
additional cost per minute for each minute or part of a minute over the
⎧
D
i
+
1
≤
D
i
≤
0
⎪
maximum
M
i
. Here
∀
i
=
1,2,.
..
N
−
1 we have
⎨
M
i
+
1
≤
M
i
≤
0 .
⎪
⎩
0
≤
C
i
+
1
≤
C
i
Problem Set #1
1
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View Full Document1.203J / 6.281J / 13.665J / 15.073J / 16.76J / ESD.216J
Logistical and Transportation Planning Methods
For plan
i
,
M
i
is called its “free time”. So, for instance, plan 1 is the least expensive to
purchase but has the least free time and the highest cost per minute of conversation
beyond free time. On the other extreme, plan
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 Fall '06
 hansman
 Probability theory, probability density function

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