Heavy tailed distributions
Risk theory
Basic quantities of interest
Edward Furman
Department of Mathematics and Statistics
York University
November 6, 2010
Edward Furman
Risk theory 4280
1 / 23
Heavy
Risk theory: loss models and risk measures, MATH 4280 3.00 F
Instructor: Edward Furman
Homework 6
1. Demonstrate that the lognormal distribution as parameterized in Klugman,
Panjer and Willmot (2008)
0.5
0.4
0.3
W)
0.2
0.]
l 2 3
Probability function for Model 3.
3;
5e-06
46-06
36-06
f(x)
26-06
1e-06
100000 200000 x 300000 400000 500000
Density function for Model 4.
0.025
0.02
0.0l5
f(X)
0.01
0.005
4 CHAPTER 2 SOLUTIONS
0.8
0.6
RX)
04
0.2
VVVI lvvv'llvvl
1 2 x 3 4 5
Distribution function for Model 3.
0.8
0.6
RX)
04
0.2
100000 200000 x 300000 400000 500000
Distribution function for Model 4.
0.8
NAME:
STUDENT NUMBER: 30 LUTI ON 3 -
MATH 4280: QUIZ FOUR, 30 minutes
Make sure to fully explain your solutions.
1. [15 marks] The variable 5 has a compound Poisson claims distribution with the follow
'NPrMB Swom: . SOLdiioNS.
MATH 4230: ourz TWO, 3o minutes
Make sure to fully explain your sotutlons. Recall that I(cr + 1) =- aI(o.r).
1. [18 marks] Recall that the PDF of a Pareto random variable is
Midterm test
MATH 4280 3.00
February 24, 2014
Given name and surname:
Student No:
Signature:
INSTRUCTIONS:
1. Please write everything in ink.
2. This exam is a closed book exam, duration 80 minutes.
3
10 CHAPTER 3 SOLUT|ONS
For Model 2, a2 = 4,000,000 1,0002 = 3,000,000, a 2 1,732.05. Hg and
,uf, are both innite so the skewness and kurtosis are not defined.
For Model 3, 02 = 2.25 .932 = 1.3851, 0 =
14 CHAPTER 3 SOLUTIONS
which cannot be correct. Recall that the numerator of the mean residual life
is E(X)E(X /\ d). However, when a g 1, the expected value is innite and
so is the mean residual life
- 1. [5 marks] Let X be a positive random variable having nite mean, the pdf f (x) and the survival function
S (w). Prove the following statements:
(i) lE[X A u] = [on S(a:)da: for u 2 0.
(ii) IE[X]
FULL NAME:
STUDENT ID:
SIGNATURE:
York University
Faculty of Science
Department of Mathematics and Statistics
MATH 4280 A
Assignment
Nov 08, 2017
INSTRUCTIONS:
This paper has 12 pages.
Total number
FULL NAME:
STUDENT ID:
SIGNATURE:
York University
Faculty of Science
Department of Mathematics and Statistics
MATH 4280 A
Assignment
October 04, 2017
INSTRUCTIONS:
This paper has 12 pages.
Total num
Actuarial models
Denition 1.10 (Univariate elliptical family)
We shall say that X E(a, b) is a univariate elliptical random
variable if when X has a density, it is given by the form
c
1 x a 2
, x