Practice Questions 3 – ACTSC 431/831, FALL 2011
1. The conditional hazard rate function of loss
X
, given Λ =
λ
, is
h
(
x

λ
) =
λx
3
. The distribution
of Λ is a gamma distribution
G
(2
,
4).
(a) Calculate the probability that the loss is less than one.
(b) Justify if the distribution of
X
is DFR, or IFR, or neither.
(c) Compare the distribution of loss
X
with that of loss
X
4
to determine whether or not one
loss has a heavier tail than the other.
2. Let loss
X
have a threecomponent spliced distribution. The pdf of
X
has the following form:
f
X
(
x
) =
0
,
x <
0
,
0
.
06
,
0
≤
x <
5
,
0
.
1
,
5
≤
x <
10
,
a
x
4
,
x
≥
10
.
(a) Calculate the variance of the loss.
(b) Calculate the distribution function
F
X
(
x
) of the loss for all
x
∈
(
∞
,
∞
).
(c) Calculate the probability that the loss is between 4 and 8.
3. Let
N
be the number of claims in an insurance portfolio. The sizes of claims in the portfolio
are i.i.d. random variables with common
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 Fall '09
 laundriualt
 Probability theory, Calculate, threecomponent spliced distribution

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