Review problems
Problem 1
Suppose
X
∼
b
(
n, p
).
a. The calculation of binomial probabilities can be computed by means of the following
recursion formula
. Verify this formula.
P
(
X
=
x
+1)=
p
(
n
−
x
)
(
x
+1)(1
−
p
)
P
(
X
=
x
)
b. Let
X
∼
b
(8
,
0
.
25). Use the above result to calculate
P
(
X
= 1), and
P
(
X
=2)
. You
are given that
P
(
X
=0)=0
.
1001
.
Problem 2
The amount of Four used per week by a bakery is a random variable
X
having an exponential
distribution with mean equal to 4 tons. The cost of the Four per week is given by
Y
=3
X
2
+1.
a. ±ind the median of
X
.
b. ±ind the 20
th
percentile of the distribution of
X
.
c. What is the variance of
X
?
d. ±ind
P
(
X>
6
/X >
2).
e. What is the expected cost?
Problem 3
Answer the folowing questions:
a. If the probabilities of having a male or female o²spring are both 0.50, ³nd the proba
bility that a family’s ³fth child is their second son.
b. Suppose the probability that a car will have a Fat tire while driving on the 405 freeway
is 0.0004. What is the probability that of 10000 cars driving on the 405 freeway fewer
than 3 will have a Fat tire. Use the Poisson approximation to binomial for faster
calculations.
c. A doctor knows from experience that 15% of the patients who are given a certain
medicine will have udesirable side e²ects. What is the probability that the tenth
patient will be the ³rst to show these side e²ects.
d. Suppose
X
follows the geometric probability distribution with
p
=0
.
2.
±ind
P
(
X
≥
10).
e. Let
X
∼
b
(
n,
0
.
4). ±ind
n
so that
P
(
X
≥
1) = 0
.
99.
11
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentProblem 4
For a certain section of a pine forest, the number of diseased trees per acre,
X
, follows the
Poisson distribution with
λ
= 10. The diseased trees are sprayed with an insecticide at a
cost of $3
.
00 per tree, plus a ±xed overhead cost for equipment rental of $50
.
00.
a. Find the probability that a randomly selected acre from this forest will contain at least
12 diseased trees.
b. Letting
C
denote the total cost for a randomly selected acre, ±nd the expected value
and standard deviation of
C
.
Problem 5
A particular sale involves 4 items randomly selected from a large lot that is known to contain
10% defectives. Let
X
denote the number of defectives among the 4 sold. The purchaser of
the items will return the defectives for repair, and the repair cost is given by
C
=3
X
2
+
X
+2.
Find the expected repair cost.
Problem 6
The telephone lines serving an airline reservation oﬃce all are busy 60% of the time.
a. If you are calling this oﬃce, what is the probability that you complete your call on the
±rst try? the second try? the third try?
b. If you and your friend must both complete calls to this oﬃce, what is the probability
that it takes a total of 4 tries for both of you to get through?
Problem 7
In the daily production of a certain kind of rope, the number of defects per foot
X
is assumed
to have a Poisson distribution with mean
λ
= 2. The pro±t per foot when the rope is sold
is given by
Y
, where
Y
=50
−
2
X
−
X
2
. Find the expected pro±t per foot.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 CRISTOU
 Binomial, Probability, Probability theory, Binomial distribution

Click to edit the document details