Dr. Valerie R. Bencivenga
Economics 329
PRACTICE HOMEWORK #6B:
CONTINUOUS RANDOM VARIABLES
1.
a.
An airline knows that the duration of the flight from Pittsburgh to Minneapolis is uniformly
distributed between 120 and 140 minutes.
The flight departs at 1:00 p.m.
The airline wants the
probability that the flight will be late not to exceed .25.
What scheduled arrival time should the
airline announce?
b.
If the flight is early, it may have to wait until its arrival gate is available for deplaning.
Assume that
the arrival gate becomes available between 3:00 p.m. and 3:10 p.m, and that the time the arrival
gate becomes available and the arrival time of the flight are
statistically independent
and
jointly
uniformly distributed.
What is the probability that the flight arriving from Pittsburgh will have to
wait for its gate?
c.
Assume the arrival gate becomes available between 3:00 p.m. and the scheduled arrival time,
and
(as in part b) the time the arrival gate becomes available and the arrival time are statistically
independent and jointly uniformly distributed.
(For example, if the scheduled arrival time is 3:07,
the arrival gate becomes available between 3:00 and 3:07.)
Suppose every minute a flight is late
costs the airline $1 thousand dollars, and every minute a flight has to wait for the arrival gate to
become available costs $2 thousand dollars.
What scheduled arrival time minimizes expected cost?
2.
Which of the following are probability density functions?
Show why or why not.
a.
otherwise
0
2
x
0
1
)
x
(
f
X
b.
otherwise
0
2
x
0
x
)
8
/
3
(
)
x
(
f
2
X
c.
otherwise
0
4
x
2
x
)
4
/
1
(
1
2
x
0
x
)
4
/
1
(
)
x
(
f
X
3.
For each part, sketch the pdf’s of the normal random variables X and Y on the same graph.
For each pdf,
label the values of the random variable corresponding to the mean and +/ one standard deviation.
Also, for
each part, state Y as a linear transformation of X.
Remember that the area under every pdf equals one!
a.
X has a mean of 5 and a variance of one; Y has a mean of zero and a variance of one.
b.
X has a mean of zero and a variance of 4; Y has a mean of zero and a variance of one.
c.
X has a mean of 5 and a variance of 4; Y has a mean of zero and a variance of one.
d.
X has a mean of 5 and a variance of 9; Y has a mean of 2 and a variance of 4.
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An investment analyst reports that the annual returns on common stocks are approximately normally
distributed with a mean return of 12.4 percentage points, and a standard deviation of 20.6 percentage
points.
On the other hand, the analyst reports that the annual returns on taxfree municipal bonds are
approximately normally distributed with a mean annual return of 5.2 percentage points, and a standard
deviation of 8.6 percentage points.
For each of the following, compute the probability of the event.
Also,
sketch the relevant normal probability density function, and indicate the area representing the computed
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 Spring '12
 BENCIVENGA
 Economics, Normal Distribution, Standard Deviation, Variance, Probability theory, probability density function

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