Practice Questions 2 – ACTSC 431/831, FALL 2011
1. Suppose that loss
Y
has a twopoint mixture distribution with the following cdf:
F
Y
(
y
) = 0
.
2
F
1
(
y
) + 0
.
8
F
2
(
y
) for all
y,
where
F
1
(
y
) and
F
2
(
y
) are the distribution functions of
1
√
X
and
e
X
, respectively. Here,
X
has a gamma distribution
G
(2
,
1
/
4). Hence,
F
1
(
y
) and
F
2
(
y
) are inverse transformed
gamma distribution and loggamma distribution, respectively.
(a) Calculate the mean of the loss.
(b) Calculate the probability that the loss exceeds its mean.
(c) Calculate the variance of the loss.
2. Let two independent random variables
X
1
and
X
2
have distributions
F
1
and
F
2
, re
spectively. Actuary A assumes that the loss of a portfolio is
Y
A
and the distribu
tion of
Y
A
is the average of the distributions
F
1
and
F
2
, namely, the cdf of
Y
A
is
F
Y
A
(
y
) =
1
2
(
F
1
(
y
) +
F
2
(
y
)) for
y
∈
(
∞
,
∞
), while actuary
B
assumes the loss of the
same portfolio is
Y
B
, which is the average of
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 Fall '09
 laundriualt
 Probability theory, yB

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