*Whenever the first word of a problem is pre
ceeded by an asterisk, the problem is for enrich
ment purposes only.
Chapter 1 Lecture Examples: FALL 2011
1. A CM has a sample space that consists of
four elements, denoted: a, b, c and d. As
suming the ELC, find the probabilities of
each of the following events.
(a)
A
=
{
a
}
(b)
B
=
{
a, b
}
(c)
C
=
{
b, c, d
}
2. Refer to the previous problem.
Now, in
stead of the ELC, assume that the probabil
ities of a, b, c and d follow the ratio 9:3:3:1.
(Note: If interested, see
Mendelian inheri
tance
in Wikipedia for a discussion of the
9:3:3:1 ratio, as well as the 1:2:1 and the
3:1 ratios.)
(a) Determine the probabilities of the in
dividual outcomes a, b, c and d.
(b) Calculate
the
probabilities
of
the
events
A
,
B
and
C
given in the pre
vious problem.
3. You are given the following information:
the events
A
and
B
are disjoint;
P
(
A
) =
0
.
40
; and
P
(
B
) = 0
.
25
. Calculate the fol
lowing probabilities.
(a)
P
(
A
or
B
)
.
(b)
P
(
A
c
)
.
(c)
P
(
B
c
)
.
4. You are given the following information:
P
(
A
) = 0
.
25
;
P
(
B
) = 0
.
45
;
P
(
AB
) =
0
.
20
. Calculate
P
(
A
or
B
)
.
5. What is wrong with each of the following?
(a)
P
(
A
) = 0
.
20
;
P
(
B
) = 0
.
55
; and
P
(
AB
) = 0
.
25
.
(b)
P
(
A
) = 0
.
60
;
P
(
B
) = 0
.
55
; and
A
and
B
are disjoint.
6. Consider a sample space with three mem
bers: 1, 2 and 3. Assume the ELC and i.i.d.
trials. The following table helps to visual
ize the results of the first two trials:
X
2
X
1
1
2
3
1
(1,1)
(1,2)
(1,3)
2
(2,1)
(2,2)
(2,3)
3
(3,1)
(3,2)
(3,3)
The nine entries in this table are equally
likely.
Define
X
=
X
1
+
X
2
, the total of the num
bers obtained in the first two trials. Find the
sampling distribution of
X
.
7. Consider a sample space with five mem
bers: 0, 1, 2, 3 and 4.
Assume the ELC
and i.i.d. trials. The following table helps
to visualize the results of the first two trials:
X
2
X
1
0
1
2
3
4
0
(0,0)
(0,1)
(0,2)
(0,3)
(0,4)
1
(0,1)
(1,1)
(1,2)
(1,3)
(1,4)
2
(0,2)
(2,1)
(2,2)
(2,3)
(2,4)
3
(0,3)
(3,1)
(3,2)
(3,3)
(3,4)
4
(0,4)
(4,1)
(4,2)
(4,3)
(4,4)
The 25 entries in this table are equally
likely.
Define
X
=
X
1
X
2
, the product of the num
bers obtained in the first two trials. Find the
sampling distribution of
X
.
1
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Chapter 1 Lecture Examples: Continued
8. Consider a sample space with three mem
bers: 1, 2 and 3. Do not assume the ELC.
Instead assume the following:
P
(1) = 0
.
2
, P
(2) = 0
.
1
and
P
(3) = 0
.
7
.
Assume i.i.d.
trials.
The following table
helps to visualize the results of the first two
trials:
X
2
X
1
1
2
3
1
(1,1)
(1,2)
(1,3)
2
(2,1)
(2,2)
(2,3)
3
(3,1)
(3,2)
(3,3)
Note that these nine entries are not equally
likely.
Define
X
=
X
1
+
X
2
, the total of the num
bers obtained in the first two trials. Find the
sampling distribution of
X
.
9. Refer to the previous question. Let
X
=
X
1
+
X
2
+
X
3
, the total of the numbers
obtained in the first three trials.
Find the
sampling distribution of
X
.
10. Consider the CM: Cast a balanced die five
times and compute the sum,
X
, of the five
numbers obtained. Assume independence
of casts. The table below presents a huge
amount of information: the exact probabil
ities for
X
; the computer simulation ap
proximations based on
m
=
100,000 runs;
the nearly certain interval for each
P
(
X
=
x
)
.
Note that every nearly certain inter
val contains the exact probability; i.e. every
one is correct.
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 Fall '11
 hanlon
 Mean, Probability theory, lecture examples

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