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ASSIGNMENT 2 – ACTSC 431/831, FALL 2008
Due at the beginning of the tutorial on Wednesday, October 8
.
1. 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
X
1
and
X
2
, namely
Y
B
=
1
2
(
X
1
+
X
2
).
(a) Assume that
F
1
is a normal distribution
N
(10
,
5) and
F
2
is a normal distribution
N
(50
,
8).
i. Calculate the means of
Y
A
and
Y
B
.
[2 marks]
ii. Calculate the variances of
Y
A
and
Y
B
.
[4 marks]
iii. Prove or disprove that
Y
A
has a normal distribution.
[8 marks]
iv. Prove or disprove that
Y
B
has a normal distribution.
[6 marks]
(b) Assume that
F
1
is an exponential distribution with mean 10 and
F
2
is an expo
nential distribution with mean 50.
i. Calculate the probability that
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This note was uploaded on 02/02/2010 for the course ACTSC 331 taught by Professor David during the Fall '09 term at Waterloo.
 Fall '09
 david

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