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STAT 230
Test
4
March 7, 2007
3:30 pm
–
4:10 pm
1. An insurance company sells a oneyear automobile policy at a fixed premium rate
R
.
The total amount of loss,
X
, incurred from a particular policy holder and paid by the
insurance company, is a random variable with possible values 0, 1, 2, 3, 4 or 5 (in
thousand dollars).
It is known that
95
.
0
)
0
(
=
=
X
P
and
x
k
x
X
P
/
)
(
=
=
for
=
x
1, 2, 3,
4 and 5, where
is a constant.
k
[4]
(a) Show that the constant
k
≈
0.022
.
Solution:
We must have
05
.
0
)
5
1
4
1
3
1
2
1
1
(
=
+
+
+
+
k
, which gives
022
.
0
≈
k
.
[4]
(b) Find the expectation
and the variance
.
)
(
X
E
)
(
X
Var
Solution:
;
Va
11
.
0
5
)
(
≈
=
k
X
E
r
(
X
)
=
5
k
(3
−
5
k
)
≈
0.318
.
[2]
(c) The net profit
is calculated as
X
R
Y
−
=
, where
R
is the fixed premium rate.
Find the expectation
and the variance
when
)
(
Y
E
)
(
Y
Var
20
.
0
=
R
.
Solution:
09
.
0
11
.
0
20
.
0
)
(
)
(
=
−
≈
−
=
X
E
R
Y
E
;
Var
(
Y
)
=
Var
(
X
)
≈
0.318.
[2] (d) Suppose the company sells a new policy with a deductible of 2. That is, the
insurance company will have zero loss (
0
=
L
) if
2
≤
X
; the company’s loss is
2
−
=
x
L
if
and
x
X
=
=
x
3, 4 or 5. Find the company’s expected loss
under the new policy,
assuming the policyholder’s loss,
)
(
L
E
X
, follows the same distribution as before.
Solution:
0315
.
0
)
2
(
)
(
5
3
≈
−
=
∑
=
x
k
x
L
E
n
.
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View Full Document2. Aidan, a construction worker who has been unemployed for last a few years, is just
employed by JBC Construction company to work on an apartment building, which is
scheduled to start
on April 1. If it rains on April 1, the work will be postponed until the
first nonraining day after April 1. It is assumed that the probability of raining on any
given day is 0.3, independent among different days. Let
X
be the number of days the
work is postponed (
X
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