Table of Common Distributions for ACTSC 431/831
1. Discrete Distributions
(a) Poisson with parameter > 0: A random variable X is said to have a Poisson distribution denoted by X P () if X has the following probability function (pf):
Prcfw_X = k =
k e
, k
ACTSC 431 - Loss Models 1
Exercise 2
1. Suppose that the ground-up loss r.v. X has a mixture of two Pareto distributions:
FX (x) = 1 a
1
1 + x
(1 a)
2
2 + x
+2
for x 0.
,
Determine the mean and the second raw moment for X.
Solution. The loss X is the mix
Actsc 431/831 (Fall 2016)
Tutorial 5
1. Given = , N has a Poisson distribution with mean . The mixing random variable
has a uniform distribution on the interval (0,5). Determine the unconditional probability
that N > 2.
2. Let N be the number of claims i
Actsc 431/831 (Fall 2016)
Tutorial 3
1. The conditional hazard rate function of loss X, given = , is h(x|) = x3 . has a gamma
distribution GAM(2, 4).
(a) Calculate the probability that the loss is less than one.
(b) Is the distribution of X DFR, IFR, or n
Actsc 431/831 (Fall 2016)
Tutorial 3
1. Solution. We first note that the model in this question is called a frailty model in that the
hazard rate function is specified by conditioning on a second random variable. In a frailty
model, the hazard rate functi
Actsc 431/831 (Fall 2016)
Tutorial 5
1. Let g denote the p.d.f. of . Then g() = 1/5 for (0, 5). Using the conditioning technique, we obtain
Z
P(N > 2)
1 P(N 6 1) = 1
=
5
P(N 6 1| = )g()d
0
Z
5
1
d
5
0
Z 5
Z 5
= 1 0.2
e d 0.2
e d
0
0
Z 5
5
5
e d
= 1 0.
Tutorial 4 Solutions
Tutorial 5
Actsc 431/831 (Fall 2016)
1. The cdf and sf of X are
x < 0,
x < 0,
0,
1,
x
F (x) = 650
, 0 6 x < 650, and S(x) = 650x
,
0 6 x < 650,
650
1,
x > 650,
0,
x > 650.
(a) The cdf of Y P is given by
FY P (y) =
(
0,
y 6 0,
F X (y+1
ACTSC431/831
Spring 2014
Homework Set II (Due June 2, 2014)
1. 75% of claims have a normal distribution with a mean of 3,000 and a variance of 1,000,000. The
remaining 25% have a normal distribution with a mean of 4,000 and a variance of 1,000,000.
Determ
ASSIGNMENT 4 ACTSC 431/831, FALL 2012
Due at the beginning of the class on Friday, November 9
1. Let N L be the number of losses. The size of the j th loss is Xj . Assume that N L , X1 , X2 , . are
independent and X1 , X2 , . have the same distribution as
ACTSC431/831
Spring 2014
Homework Set I (Due May 21, 2014)
1. For two random variables X and Y , eY (30) = ex (30) + 4. Let X have a uniform distribution
on the interval from 0 to 100 and let Y have a uniform distribution on the interval from 0 to
w. Dete
ACTSC431/831
Spring 2014
Homework Set III (Due June 16, 2014)
1. For i = 1, . . . , n let Si have independent compound Poisson frequency distributions with
Poisson parameter i and a secondary distribution with pgf P2 (z). Note that all n of the
variables
ACTSC431/831
Spring 2014
Homework Set IV (Due July 2, 2014)
1. Losses have a Pareto distribution with = 2 and = k. there is an ordinary deductible of
2k. Determine the loss elimination ratio before and after 100% ination.
2. Losses have a mean of 2,000. W
ACTSC 431/831 - Loss Models I
Lecture 1
May 5, 2014
Lecture 1,
ACTSC 431/831 - Loss Models I
1/20
Course Information
Instructor: Dr. Xiaoying Han
Oce: 4101 M3
E-mail: x62han@uwaterloo.ca
Course Web: UW-LEARN
Lecture: M 4:00p - 5:20p @ 1350 DC
W 4:00p - 5:
ACTSC 431/831 - Loss Models I
Lecture 2
May 7, 2014
Lecture 2,
ACTSC 431/831 - Loss Models I
1/10
Examples
Ex2.1 X4 - the total dollars in medical malpractice claims paid in
one year owing to events at a randomly selected hospital:
F4 (x) =
0,
x <0
0.0000
ACTSC 431/831 - Loss Models I
Lecture 5
May 21, 2014
Lecture 5,
ACTSC 431/831 - Loss Models I
1/7
Creating New Distributions
Given distributions of a continuous random variable X, seek
distributions of X
multiplied by a constant: Y = X
raised to a power
LOSS MODEL 1ACTSC 431/831, Spring 2014
Instructor:
Lectures:
Tutorials:
Oce hours:
Dr. Chengguo Weng, M3 3136, ext.31132, c2weng@uwaterloo.ca
1:00-02:20pm Wednesday and Friday, EV3 1408.
Scheduled every another week, in MC 1085, at 8:30-9:20am; see Table
Chapter 1. Introduction and Overview
Course objective: As the name of the course suggests, it is to introduce various mathematical
models which can be used by insurers to forecast and predict future insurance losses (or risks),
primarily in a non-life ins
Practice Questions - Set 3 Solution
1. This years ground-up loss r.v. X has a distribution given by an equal mixture of two exponential
distributions, one with a mean of 250 and the other with a mean of 500.
(a) Assume an ordinary deductible of 150.
i. Fi
Actsc 431 - Term Test 1 Info
Your rst term test will be held on Monday October 7 from 3:30pm to 4:20pm (50 minutes
long).
It will cover the material up to and including page 23 of the online notes (Please also see
the document I recently emailed you on
Actsc 431: Term Test 1 - Fall 2013
Department of Statistics and Actuarial Science, University of Waterloo
October 7, 2013
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First name:
I.D.#:
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Actsc 431: Term Test 2 (Version 1) - Spring 2013
Department of Statistics and Actuarial Science, University of Waterloo
June 25, 2013
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Actsc 431: Term Test 2 - Fall 2013
Department of Statistics and Actuarial Science, University of Waterloo
October 28, 2013
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Actsc 431: Term Test 1 (Version 1) - Spring 2013
Department of Statistics and Actuarial Science, University of Waterloo
June 3, 2013
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Example 13: Suppose Y | = EXP
distribution of Y .
1
and let GAM (, ). Find the unconditional
Given the conditional distribution of Y given = , we have
SY | (y |) = P (Y > y | = ) = ey
Now, the unconditional survival function can be obtained by averaging o
ACTSC 431/831 - Convolution Method
1. Suppose N BIN (3, 0.6), and M is such that P (M = 0) = 0.2 and P (M = 4) = 0.8. Let
S = M1 + M2 + + MN and fk n = P (S = k |N = n). Then
fk n
k=0
k=4
k=8
k = 12
n=0
1
0
0
0
3
0
(0.6)0 (0.4)3
n=1
0.2
0.8
0
0
3
1
(0.6)1
Question of the Day - June 18
Due on June 19, 2013 by 4:00pm - Submit to Dropbox on Learn
Assume
S1 has a compound Poisson distribution with parameter 4 and a BIN (1, p) secondary
distribution, and
S2 has a compound Poisson distribution with parameter 3
Actsc 431 Notes
1
Introduction
In insurance loss modelling, two popular models in the literature include the individual risk
model and the collective risk model.
Individual risk model: Here, we consider the total claim amount, S on n policies/insureds.
I
Practice Questions - Set 2
1. Suppose that X | = EXP () and has p.d.f.
f () =
p1
, 0 < < , v > 0, 0 < p < 1.
(p)(1 p)( )p
(a) Show that X GAM (p, ).
(b) We have seen that the gamma distribution is a scale distribution. Show that is a
scale parameter.
(c)
Actsc 431 Notes
1
Introduction
In insurance loss modelling, two popular models in the literature include the individual risk
model and the collective risk model.
Individual risk model: Here, we consider the total claim amount, S on n policies/insureds.
I
Actsc 431: Term Test 1 (Version 1) - Spring 2013
Department of Statistics and Actuarial Science, University of Waterloo
June 3, 2013
Last name:
First name:
I.D.#:
Notes:
Show all work.
Aid: Calculator (Financial or Scientic)
Unless otherwise specied, n