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THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT1301 Probability and Statistics I
Assignment 2 Solution
1. Sample space:
{
}
BO
BB
WO
WB
WW
,
,
,
,
=
Ω
Note that the outcomes are not equally likely with associated probabilities:
{
}
(
)
3
1
45
15
2
10
2
6
=
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
WW
P
,
{
}
(
)
5
2
45
3
6
=
×
=
WB
P
{
}
(
)
15
2
45
1
6
=
×
=
WO
P
,
{
}
(
)
15
1
45
2
3
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
BB
P
,
{
}
(
)
15
1
45
1
3
=
×
=
BO
P
The random variable
X
is the following function:
(
)
2
=
WW
X
,
(
)
1
=
WB
X
,
(
)
0
=
WO
X
,
(
)
0
=
BB
X
,
(
)
1
−
=
BO
X
(a) The the possible values of
X
are
–1, 0, 1, 2.
(b) The probability mass function is given by
(
)
{
}
(
)
15
1
1
=
=
−
=
BO
P
X
P
(
)
{
}
(
)
5
1
15
1
15
2
,
0
=
+
=
=
=
BB
WO
P
X
P
(
)
{
}
(
)
5
2
1
=
=
=
WB
P
X
P
(
)
{
}
(
)
3
1
2
=
=
=
WW
P
X
P
The cumulative distribution function is
( )
(
)
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
≥
<
≤
<
≤
<
≤
−
−
<
=
≤
=
2
if
1
2
1
if
3
2
1
0
if
15
4
0
1
if
15
1
1
if
0
x
x
x
x
x
x
X
P
x
F
.
2.
( )
⎪
⎪
⎩
⎪
⎪
⎨
⎧
=
=
=
=
=
12
,
6
9
1
4
12
1
30
,
24
,
20
,
18
,
15
,
10
,
8
,
5
,
3
,
2
18
1
36
,
25
,
16
,
9
,
1
36
1
i
i
i
i
i
p

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