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THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT1801 Probability and Statistics: Foundation of Actuarial Science
Assignment 3 Solution
1. (a)
3
2
(b)
3
2
2. (a)
(b)
(
)
1
,
a
U
(
)
a
U
,
0
3.
(a) From the definition of
, we know that
1
−
F
( )
(
)
u
u
F
F
≥
−
1
for all
. Since
F
is a
non-decreasing function, we have
(
1
,
0
∈
u
)
( )
( )
(
)
( )
( )
x
F
u
x
F
u
F
F
x
u
F
≤
⇒
≤
⇒
≤
−
−
1
1
(as
( )
(
)
u
u
F
F
≥
−
1
)
On the other hand, since
( )
u
F
1
−
is the minimum
x
such that
, therefore if
then
( )
u
x
F
≥
( )
u
x
F
≥
x
must be greater than or equal to
( )
u
F
1
−
, i.e.
( )
( )
u
F
x
F
1
−
≥
x
u
⇒
≥
.
(b) The distribution function of
X
is given by
( )
(
)
(
)
(
)
( )
(
)
( )
x
F
x
F
U
P
x
U
F
P
x
X
P
x
F
X
=
≤
=
≤
=
≤
=
−
1
.
(c) Distribution function of
( )
λ
Exp
is
( )
x
e
x
F
λ
−
−
=
1
,
. Therefore
0
>
x
( )
{
}
(
)
(
)
u
u
x
x
u
e
x
u
F
x
x
x
−
−
=
⎭
⎬
⎫
⎩
⎨
⎧
−
−
≥
=
≥
−
=
−
−
1
log
1
1
log
1
:
min
1
:
min
1
λ
λ
λ
Procedure to generate a random variable from
( )
λ
Exp
:
[1]
Generate
(
)
.
1
,
0
~
U
U
[2]
Compute
(
)
U
X
−
−
=
1
log
1
λ
.
4. 97
5. 0.1719
6.
5
.
0
>
p
7. Let
X
be the number of times the stock increases in
n
weeks. Then
. Denote $
Y
as the price of the stock after
n
weeks, then
(
55
.
0
,
~
n
b
X
)
(
)
(
)
(
)
(
) (
)
X
n
X
n
X
n
X
Y
5625
.
1
8
.
0
10
8
.
0
25
.
1
8
.
0
10
8
.
0
25
.
1
10
×
=
⎟
⎠
⎞
⎜
⎝
⎛
=
×
=
−
.

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