Bridges
Math 115b
Name ____________________
April 27, 2004
Test 2B
1.
X is a continuous random variable that takes values in the interval [0, 4]. X has p.d.f.,
f X ( x)  0.09375x 2  0.375x
p.d.f.
0.4
0.3
f(x) 0.2
0.1
0
0
2
4
6
x
(a)
Set up, but do not evaluate, an integral that represents P(2.1 < X).

(
(
(
(b)
)dx
)
Set up, but do not evaluate, an integral that could be evaluated by Integrating.xls for the
expected value,  X , of X.

(
2.
)
(
)
(
)dx
)
Your company’s computer system on average experiences 22.4 hours between system crashes.
Let T be the continuous exponential random variable giving the time, in hours and fractions of
hours, between such crashes. To 4 decimal places, compute the probability that between 26 and
29 hours passes between crashes. Show all work.
1
3.
X is a finite random variable whose p.m.f. is given below.
x
f X (x)
0
0.1
4
0.6
8
0.3
To the nearest hundredth (2 decimal places), compute (a) the mean, (b) the variance, and (c) the
standard deviation of X. Show all work.
4.
Your friend scores 67 points on the Math 115b test. He wants to assure his advisor that this is a
decent score based on the scores of the other students. So, he asks six friends in class what they
scored on this test. He calculates that the mean of these six scores is 71.4 and that the standard
deviation is 7.6.
(a)
Calculate the 98% confidence level. For this confidence level, z0 = 2.326.
(b)
Explain what this confidence level means. In your explanation, tell how the student could
use the results to convince the advisor that his score is okay.
2
5.
Write the following equation in Excel beginning with the “=” symbol. Be sure to clearly include
all necessary symbols.
f X ( x) 
6.
1
2.8 2
e
 x 35.1 
0.5

 2.8 
2
The graph of the p.d.f., f X (x) , for a continuous random variable, X, is shown below.
p.d.f. of f(x)
0.14
0.12
0.1
0.08
f(x)
0.06
0.04
0.02
0
-0.02 0
2
4
6
8
10 12 14 16 18 20 22 24 26
x
(a)
Give the best possible estimate of the mean,  X , of X.
(b)
Give the best possible estimate of the standard deviation,  X , of X.
3
7.
An inspector checks plant conditions once during his shift of 12 hours that begins at 7:00 a.m.
and ends at 7:00 p.m. Let Y be the time (in hours) from the start of his shift until the inspection
starts. Y is uniform on [0, 12].
(a)
Draw a neat sketch of the p.d.f. of Y. You must label units on your axes.
y
6
x
-1
-1
13
(b)
What is the expected time of the inspection? Show all work or explain how you arrived at
your answer.
(c)
If the standard deviation of Y is 2.89, find the standardization S of Y.
(d)
To 4 decimal places, compute P(-1 < S < 1.5)
4
8.
You represent one of 8 companies that bid on oil leases. Historical data indicates that there will
be a winner’s curse of 18.6 million dollars in the auction process. Simulation shows a stable Nash
equilibrium strategy of a 20.2 million dollar signal reduction.
(a)
Suppose that all companies, yours included, bid their geologists’ estimates for the fair
value of a lease. If you win the bid, on average how much profit (positive or negative)
would your company receive on the lease and why?
(b)
Suppose that all companies, yours included, bid their geologists’ estimates for the fair
value of a lease minus 20.2 million dollars. If you win the bid, on average how much
profit (positive or negative) would your company receive on the lease and why?
(c)
What does an equilibrium strategy mean?
5