Practice Questions 1 – ACTSC 431/831, FALL 2011
1. Let
Y
be the total loss in an insurance portfolio and
N
be the number of claims in the
portfolio. If
N
= 0, then
Y
= 0. If 1
≤
N
≤
5, then
Y
has an exponential distribution
with mean 100. If
N >
5, then
Y
has an exponential distribution with mean 200. It
is given that Pr
{
N
= 0
}
= 0
.
3
,
Pr
{
1
≤
N
≤
5
}
= 0
.
5
,
and Pr
{
N >
5
}
= 0
.
2
.
(a) Let
F
Y
(
y
) be the distribution function of
Y
. Calculate
F
Y
(
y
) for all
y
∈
(
∞
,
∞
).
(b) Calculate the probability that the total loss in the portfolio will be greater than
500.
(c) Calculate the probability of Pr
{
0
< Y <
300
}
.
(d) Calculate the mean of the total loss.
(e) Calculate the 20th percentile of the total loss.
(f) Calculate the 80th percentile of the total loss.
2. The groundup loss
X
for an insurance policy has a uniform distribution
U
(0
,
200). If
X
≤
80, the policy pays nothing; if 80
< X
≤
120, the policy pays the amount of
the loss above 80; if 120
< X
≤
160, the policy pays 40 plus 75% of the loss above
120; if
X >
160, the policy pays 70. Thus, the per loss random variable for the policy,
denoted by
Y
L
, is
Y
L
=
0
,
X
≤
80;
X

80
,
80
< X
≤
120;
40 + 0
.
75(
X

120)
,
120
< X
≤
160;
70
,
X >
160
.
Let
Y
P
be the per payment random variable for the policy. Then
Y
P
=
Y
L

X >
80.
(a) Determine
F
Y
L
(
y
), the cdf of
Y
L
, for all
y
∈
(
∞
,
∞
).
(b) Determine
F
Y
P
(
y
), the cdf of
Y
P
, for all
y
∈
(
∞
,
∞
).
(c) Calculate the probability of Pr
{
40
≤
Y
L
≤
50
}
.
(d) Calculate the probability of Pr
{
30
≤
Y
P
<
70
}
.
(e) Calculate the mean of the per loss of the policy.
(f) Determine the median of the per payment of the policy.
(g) Determine the 90th percentile of the per payment of the policy.
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 Fall '09
 laundriualt
 Normal Distribution, Probability theory, 75%, Calculate, distribution function Fy

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