STAT 3810 Assignment 1 Suggestted Solutions
4
March 3, 2011
Problem 1:
(a). Since SX (x) = P (X > x) by the denition of survival function,
SX (x)dx =
P (X > x)dx
=
fX (t)dtdx
0
x
t
=
fX (t)dxdt
0 0
=
t fX (t)dt
0
0
0
= E[X]
(b). Since
fe (x)dx =
0
0
STAT3810/3906 Assignment 3
Due: April 14, 2014
Problem 1 You are given that
Total claims for a health plan have a Pareto distribution with = 2 and = 500.
The health plan implements an incentive to physicians that will pay a bonus of 50% of
the amount by
STAT3810/3906 Assignment 1
Due: March 3, 2014
Problem 1 Let X be a positive random variable with probability density function fX (x), x 0.
(a) Show that the function
fe (x) :=
SX (x)
,
E[X]
x0
denes a probability density function.
(b) Let Y be a positive
STAT3810/3906 Assignment 2
Due: March 22, 2014
Problem 1 Suppose that the loss X has a Pareto distribution with = 2 and = 100 (refer
to the appendix of the textbook for the parametrization of the Pareto distribution).
(a) Determine the range of the mean e
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3810
Risk Theory
Spring 2014 Tutorial 6
1. Let X be the towing cost per tow, N be the number of tows, S be the aggregate towing
cost. We have
N
S=
Xi .
i=1
Note that
E (S) =
=
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3810/3906
Risk Theory
Spring 2014 Tutorial 7 Solution
1. For an Exponential distribution with mean ,
d
ex/
ex/
dx +
dx
d
E (X d) =
x
d
0
= ded/ + 1 ed/ + ded/
= 1 ed/
E (X d
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3810/3906
Risk Theory
Spring 2014 Tutorial 7
1. You are given:
Losses follow and exponential distribution with the same mean in all years.
The loss elimination ratio this yea
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3810
Risk Theory
Spring 2014 Tutorial 5
1. (a) Let pT = Pr X T = k . Then,
k
E X
T
i Pr X T = i
=
i=1
i
=
i=1
i=0
Pr (X = i)
1 Pr (X = 0)
i Pr (X = i)
1 Pr (X = 0)
E (X)
=
1 p0
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3810
Risk Theory
Spring 2014 Tutorial 5
1. Let pk = Pr (X = k), where X is a discrete random variable taking values
0, 1, 2, . . .
(a) Let X T be the zero-truncated random vari
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3810
Risk Theory
Spring 2014 Tutorial 6
1. A towing company provides all towing services to members of the City Automible
Club. You are given:
Towing Distance Towing Cost Frequ
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3810
Risk Theory
Spring 2013 Tutorial 4 Solution
1. Denote N the annual number of accidents. Then,
PN (t) = E tN
= E E tN |
= E e(t1)
= M (t 1)
= [1 (t 1)] ,
which is the p.g.f
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3810/3906
Risk Theory
Spring 2014 Tutorial 3 Solution
1. By double expectation formula, we have
E (W ) = E 2N
= E E 2N |
= E PN | (2)
= E MN | (ln 2)
ln 2
= E e(e 1)
( MX (ln t
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3810/3906
Risk Theory
Spring 2014 Tutorial 3
1. Bob is a carnival operator of a game in which a player receives a prize worth W = 2N
if the player has N successes, N = 0, 1, 2,
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3810
Risk Theory
Spring 2014 Tutorial 4
1. The annual number of accidents for an individual driver has a Poisson distribution with
mean . The poisson mean, , has a gamma distri
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3810/3906
Risk Theory
Spring 2014 Tutorial 2
1. (a) Suppose that, conditional on = , X follows a normal distribution with mean
and variance 2 , where follows another normal di
1
fY (y) =
0
,0 < y <
,
fY (y + d)
SY (d)
1/
=
d
1
1
,0 < y < d
d
=
0
,
fY d|Y >d (y) =
eY (d) = E (Y d |Y > d)
d
y
=
dy
d
0
d
=
2
eY (30) = eX (30) + 4
100 30
30
=
+4
2
2
= 108
E (X u) =
u
SX (x) dx
0
0
FX (x) dx
1000
E
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3810/3906
Risk Theory
Spring 2014 Tutorial 2 Solution
1. (a) We make use of the moment generating function technique to obtain its distribution.
In particular,
MX (t) = E etX
=
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3810/3906
Risk Theory
Spring 2014 Tutorial 1
Tutor: Mr. Wong Tsun Yu, Je
Oce: MW515
Tel: 2859-1920
Email: siuyu88@hku.hk
Tutor: Mr. Liu Haibo, Paul
Oce: MW515
Tel: 2859-1920
Em
STAT3810/3906
Chapter 5 Frequency Models - III
by K.C. Cheung
Learning Objectives:
Understand the idea of compound frequency model
Compute moments and PGF of a compound frequency distribution
Understand the convolution property of compound Poisson dist
STAT3810/3906
Chapter 2 Technique of Mixing
by K.C. Cheung
Learning Objectives:
Apply the technique of mixing to create new distributions
Understand the meaning of a frailty model
2.1 Mixing
2.1.1 Suppose that X is a random variable with density functio
STAT3810/3906
Chapter 4 Frequency Models - II
by K.C. Cheung
Learning Objectives:
Frequency model: the (a, b, 0) class
Truncation and modication at zero: the (a, b, 1) class
4.1 The (a, b, 0) Class
4.1.1 Let pk = P(N = k), where N is a discrete random v
STAT3810/3906
Chapter 3 Appendix - A Deeper Look at the NB
Distribution
by K.C. Cheung
3A.1 Probability Function
3A.1.1 Recall: If N is a negative binomial random variable with parameters r > 0 and > 0,
then
r
k
k+r1
1
,
k = 0, 1, 2, . . .
pk = P(N = k) =
STAT3810 (2013-2014)
Chapter 1 Basic Distributional Quantities
by K.C. Cheung
Calculate some basic distributional quantities:
moments, percentiles, generating functions
Understand the meaning of:
excess loss variable, stop loss variable, limited loss