Tutorial 7
1. Let X and Y have joint probability distribution as given in Table 20.6.
Table 20.6 Data for Exercise 20.20
y
x
0
1
2
0
0.20
0
0.10
1
0
0.15
0.25
2
0.05
0.15
0.10
(a) Compute the marginal distributions of X and Y .
(b) Compute the conditional
Tutorial 1
1. Let X have a Pareto distribution with parameters and . Let Y = ln(1 + X ).
Determine the name of the distribution of Y and its parameters.
Solution. The distribution function of Y is
FY (y ) = P(Y y )
X
)
= P(ln(1 +
= P(1 +
X
y)
ey )
= P(X
Tutorial 10
1. Suppose that, given = , the random variables X1 , X2 , , Xn are independent with Poisson pdf
fXj | (xj |) =
xj e
,
xj !
xj = 0, 1, 2, .
(a) Let S = X1 + + Xn . Show that S has pdf
fS (s) =
0
(n)s en
()d,
s!
s = 0, 1, 2, ,
where has pdf ().
Tutorial 8
1. Suppose that, given = , the random variables X1 , X2 , , Xn are independent with Poisson pdf
fXj | (xj |) =
xj e
,
xj !
xj = 0, 1, 2, .
(a) Let S = X1 + + Xn . Show that S has pdf
fS (s) =
0
(n)s en
()d,
s!
s = 0, 1, 2, ,
where has pdf ().
Tutorial 9
1. Three urns have balls marked 0, 1 and 2 in the proportions given in Table 1.
An urn is selected at random, and two balls are drawn from that urn with
replacement. A total of 2 on the two balls is observed. Two more balls are then
drawn with
Tutorial 4
1. Suppose that given = the random variables X1 , , Xn are independent
and identically exponentially distributed with pdf
fXj | (xj |) = exj ,
xj > 0,
and is itself gamma distributed with parameters > 1 and > 0,
1 e
() =
,
()
> 0.
(a) Verif
Tutorial 5
1. An insurance company has decided to establish its full-credibility requirements
for an individual state rate ling. The full-credibility standard is to be set so
that the observed total amount of claims underlying the rate ling would be
withi
Tutorial 2
1. The distribution of S = X1 + X2 + + XN , when N is random, is called a
compound distribution. What is the moment generating function of S ?
Solution. The moment generating function of S is easier to derive. Let MX (t)
be the mgf of X , and M
Tutorial 3
1. The claim frequency N has a negative binomial distribution with r = 2 and
= 1.5. The severity X takes values 20, 40, 60, 80 or 100 with equal probability.
(a) If the insurer imposes a per-claim deductible of 20, what is the expected
aggrega
Tutorial 11
1. Suppose that there are two types of policyholder: type A and type B. Twothirds of the total number of the policyholders are of type A and one-third are
of type B. For each type, the information on annual claim numbers and severity
arc given
Assignments 1
Due date: October 22, 2013
Mid-term test time: From 19:00 to 21:00, October 29, 2013
Q20.1. An insurance company has decided to establish its full-credibility requirements for
an individual state rate ling. The full-credibility standard is t
Assignments 2
Due date: December 3, 2013
Q20.32. Suppose that X1 , X2 , , Xn are independent (conditional on ) and
E[Xj |] = j ()
and
Var(Xj |) =
2j v ()
,
mj
j = 1, , n.
Let = E[()], v = E[v ()], a = Var(), k = v/a and m = m1 + m2 + mn .
(a) Discuss whe