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Homework 1 – solution
1. Either company needs to adjust its premium so that it can make
payoﬀs for up to
x
*
ships without losing money, where
x
*
is the smallest
number such that
P
(
X > x
*
)
<
0
.
05 (where
X
∼
Bin
(
n,
0
.
015) is the
number of lost ships). Using excel, it’s easy to ﬁnd that, for the smaller
company (
n
= 1000),
x
*
= 22, and for the larger company (
n
= 2000),
x
*
= 39. So the minimum premium that allows to satisfy the requirement of
risk being no more than 0.05 is:
22
·
100000
1000
= 2200, for the smaller company;
and
39
·
100000
2000
= 1950, for the larger company.
2. The lifetime of any given bulb is
T
∼
N
(10000
,
1000).
a)
P
(
T >
12000) = 1

F
(12000) = 1

Φ
‡
12000

10000
1000
·
= 1

Φ(2) =
1

0
.
977 = 0
.
023.
b)
P
(
T <
9000) =
F
(9000) = Φ
‡
9000

10000
1000
·
= Φ(

1) = 0
.
159.
c)
P
(replacing at least one bulb) = 1

P
(replacing no bulbs) = 1

P
(all
T
i
>
5000) = 1

P
(
T >
5000)
100
= 1

(1

F
(5000))
100
= 1

(1

Φ(

5))
100
= 0 (up to three decimal points).
d)
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 Spring '08
 Perevalov

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