York University
MATH 2030 3.0 (Elementary Probability) Fall 2011
Assignment 4 – Solutions, Oct 2011
§
3.1 No. 9
X
= 2 if the 2nd draw matches the first, which has probability
1
7
.
X
= 3 if the
2nd draw differs from the first, and the third matches the 1st or 2nd. This has
probability
6
7
×
2
6
=
2
7
. Likewise
P
(
X
= 4) =
6
7
×
4
6
×
3
5
=
12
35
. If the first 4 draws
differ, then the 5th must match one of them, so
P
(
X
= 5) = 1

1
7

2
7

12
35
=
8
35
.
We obtain
x
2
3
4
5
P
(
X
=
x
)
1
7
2
7
12
35
8
35
§
4.1 No. 3abcd
(a) We want
1 =
∞
∞
f
(
x
)
dx
=
1
0
cx
(1

x
)
dx
=
c
1
0
(
x

x
2
)
dx
=
c
x
2
2

x
3
3
1
0
=
c
(
1
2

1
3
) =
c
6
So we take
c
= 6.
(b) Using part (a),
P
(
X
≤
1
2
) =
1
/
2
∞
f
(
x
)
dx
=
1
/
2
0
6
x
(1

x
)
dx
=
1
/
2
0
(6
x

6
x
2
)
dx
=
3
x
2

2
x
3
1
/
2
0
=
3
4

1
4
=
1
2
(c)
P
(
X
≤
1
3
) =
1
/
3
∞
f
(
x
)
dx
=
1
/
3
0
6
x
(1

x
)
dx
=
1
/
3
0
(6
x

6
x
2
)
dx
=
3
x
2

2
x
3
1
/
3
0
=
1
3

2
27
=
7
27
= 0
.
2593
(d)
P
(
1
3
< X
≤
1
2
) =
P
(
X
≤
1
2
)

P
(
X
≤
1
3
) =
1
2

7
27
=
13
54
= 0
.
2407
§
4.5 No. 2a
Let
X
∼
Bin(3
,
1
2
). Using the formula
P
(
X
=
k
) =
(
3
k
)
(1
/
2)
k
(1
/
2)
3

k
we compute
that
P
(
X
= 0) = 1
/
8,
P
(
X
= 1) = 3
/
8,
P
(
X
= 2) = 3
/
8, and
P
(
X
= 3) = 1
/
8.
So the cdf
F
(
x
) of
X
is a step function.
F
(
x
) = 0
,
x <
0
,
;
F
(
x
) = 1
/
8
,
x
∈
[0
,
1);
F
(
x
) = 1
/
8 + 3
/
8 = 4
/
8
,
x
∈
[1
,
2);
F
(
x
) = 4
/
8 + 3
/
8 = 7
/
8
,
x
∈
[2
,
3);
F
(
x
) = 7
/
8 + 1
/
8 = 1
,
x
≥
3. You draw a pic according to this. Note that the
1
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2
jump height at each jump point
x
= 0
,
1
,
2
,
3 follows from the defining pr at each
x
= 0
,
1
,
2
,
3.
Note that at the jump points
x
= 0
,
1
,
2
,
3 the cdf takes the upper value, so that
the function is rightcontinuous with leftlimits (this is the defining property of a
cdf).
§
4.5 No. 5
F
(
x
) =
P
(
X
≤
x
) =
x
∞
f
(
t
)
dt
=
x
∞
e
t
2
dt,
x <
0
0
∞
e
t
2
dt
+
x
0
e

t
2
dt,
x
≥
0
=
x
∞
e
t
2
,
x <
0
0
∞
e
t
2
+
x
0

e

t
2
,
x
≥
0
=
e
x
2
,
x <
0
1

e

x
2
,
x
≥
0
§
4.5 No. 6ab
(a) Because
P
(
X
= 1
/
2) = 0 we have that
P
(
X
≥
1
/
2) = 1

P
(
X <
1
/
2) =
1

F
(1
/
2) = 1

1
8
= 7
/
8.
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 Probability, Probability theory, Binomial distribution, dx, No., rolls

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