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AMS 311 (Fall, 2010)
Joe Mitchell
PROBABILITY THEORY
Homework Set # 5 – Solution Notes
(1).
(15 points)
Ten years ago at a certain insurance company, the size of claims under homeowner insurance policies
had an exponential distribution. Furthermore, 25% of claims were less than
$
1000. Today, the size of claims still
has an exponential distribution but, owing to in±ation, every claim made today is twice the size of a similar claim
made 10 years ago. Determine the probability that a claim made today is less than
$
1000.
Let
X
be the size (in dollars) of a claim ten years ago; we know
X
is exponential(
λ
). We are told that
P
(
X <
1000) = 0
.
25, so we know that 1

e

1000
λ
= 0
.
25, so
e

1000
λ
= 3
/
4. (This tells us that
λ
=
ln(0
.
75)

1000
.)
Let
Y
be the size (in dollars) of a claim today. We are told that
Y
= 2
X
.
We want
P
(
Y <
1000) =
P
(2
X <
1000) =
P
(
X <
500) = 1

e

500
λ
= 1

√
e

1000
λ
= 1

r
3
/
4 = 1

(
√
3
/
2)
≈
0
.
13397
.
(2).
(15 points)
In a particular forest, the distance between any randomly selected tree and the tree nearest to it is
exponentially distributed with a mean of 40 feet. (a). Find the probability that the distance from a randomly selected
tree to the tree nearest to it is more than 30 feet. (b). Find the probability that the distance from a randomly selected
tree to the tree nearest to it is more than 80 feet, given that the distance is at least 50 feet. (c). Find the minimum
distance that separates at least 50% of the trees from their nearest neighbor.
Consider a randomly selected tree, and let
X
denote the distance, in feet, from this tree to its nearest neighbor.
We are told that
X
is exponential(
λ
), with
E
(
X
) = 40; thus,
λ
= 1
/
40.
(a). We want
P
(
X >
30) =
i
∞
30
(1
/
40)
e

(1
/
40)
x
dx
=
e

3
/
4
.
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This note was uploaded on 07/15/2011 for the course AMS 311 taught by Professor Tucker,a during the Spring '08 term at SUNY Stony Brook.
 Spring '08
 Tucker,A

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