Part VI: Simulation
Chapter 20: Simulation
The goal of simulation is to determine values relating to the
distribution of a random variable 8:
0 Build a model for S which depends on random variables X, Y, Z, 5: S (Y , X Z ~ >
., Where their distributions a
Midterm Exam - ACT 4630
2:30 PM 4:30 PM, Thursday, March 9, 2017
(
Name (print): 30 (UL-'1 Oh Score:
Write legibly and show your steps!
1. Michael is a professional stuntman Who performs dangerous motorcycle jumps at ex-
treme sports events around the wor
Simulation
Two Rule For Inverting A Distribution Function
Rule 1. If F(x) = u is constant on an interval [x1 , x2 ), then the uniform value u is mapped onto x2
through the inversion process. (this is said on the last paragraph of page 440 of the textbook)
Limited Fluctuation Credibility
Limited Fluctuation Credibility (also called the classical approach):
Update the prediction of loss, as a weighted average of the prediction based on recent data and the rate
taken from the insurance manual.
Full Credibili
Simulation
Two Rule For Inverting A Distribution Function
Rule 1. If F(x) = u is constant on an interval [x1 , x2 ), then the uniform value u is mapped onto x2
through the inversion process. (this is said on the last paragraph of page 440 of the textbook)
Appendix
Definition 2.1
The cumulative distribution function (cdf): FX (x) = Pr(X x).
Definition 2.4
The survival function: SX (x) = Pr(X > x) = 1 FX (x).
Definition 2.5
0 (x ) = S0 (x ).
The probability density function (pdf): fX (x) = FX
X
Definition 2.
Chapter 5: Continuous models
5.2 Creating new distributions
(i) Multiplication by a constant
Theorem 5.1
Let X be a continuous random variable with pdf fX (x) and cdf FX (x).
Let Y = X with > 0. Then,
y
FY (y) = FX ( ),
Samuel Hao (University of Manitoba)
Chapter 10: Review of mathematical statistics
10.2 Point estimation
Population random variable: X
Sample: X1 , X2 , . . . , Xn (i.i.d. copies of X)
Estimator: = (X1 , . . . , Xn )
Definition 10.1
is unbiased if E | = for all . The bias is
An estimator, ,
Chapter 8: Frequency and severity with coverage
modifications
8.2 Deductibles
Ordinary deductible
Samuel Hao (University of Manitoba)
Franchise deductible
ACT 4630
Winter 2017
40 / 284
Theorem 8.3
For an ordinary deductible, the expected cost per loss is
Chapter 13: Frequentist Estimation
Suppose we have n observations from a parametric distribution
F(x| )
with = (1 , . . . , p )T .
Here are two basic methods for estimating :
Definition 13.1
A method-of-moments estimate of is any solution of the p equatio
Chapter 12: Estimation for modified data
12.1 Point estimation
Definition 12.1
An observation is truncated from below (also called left truncated) at
d if when it is below d it is not recorded but when it is above d it is
recorded at its observed value.
A
Chapter 11: Estimation for complete data
Data Set A This data set is well-known in the casualty actuarial
literature. It collects data from 19561958 on the number of accidents
by one driver in one year.
Data Set A.
Number of accidents
Number of drivers
0
ACT 4630: Construction and Evaluation of Actuarial
Models
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University of Manitoba
Winter 2017
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ACT 4630
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Chapter 3: Basic distributional quantities
3.4 Tails of distributions
Samuel Hao (Univ
Chapter 9: Aggregate loss models
9.3 The compound model for aggregate claims
Definition 9.1
The collective risk model has the representation
S = X1 + X2 + + XN ,
where Xj s are individual payment amounts, and N is the number of
payments. This is a compoun
16.5 Selecting a model
In selecting models, there are two things to keep in mind:
1
Use a simple model if at all possible (the principle of parsimony).
2
Restrict the universe of potential models.
Samuel Hao (University of Manitoba)
ACT 4630
Winter 2017
1
9.6 The recursive method
Suppose
Severity distribution fX (x) is defined on 0, 1, 2, . . . , m representing
multiples of some convenient monetary unit.
Frequency distribution pk , k = 0, 1, . . ., is from (a, b, 1) class.
Theorem 9.8
For x = 0, 1, 2, . .
12.3 Kernel density models
Let p(yj ) be the probability assigned to the value yj for j = 1, . . . , n by
the empirical distribution. Let Kj (x) be a distribution function for a
continuous distribution such that its mean is yj . Let kj (x) be the
correspo
Example 12.20
Estimate single-decrement probabilities using Data Set D2 and the
actuarial method. Making reasonable assumptions.
Table 12.9 (revised)
j
0
1
2
3
4
nbj
30
0
0
0
0
Pj
30
29
28
26
23
Samuel Hao (University of Manitoba)
Single-decrement mortali
Exam MLC
Adapt to Your Exam
SURVIVAL DISTRIBUTIONS
SURVIVAL DISTRIBUTIONS
Moments
Complete Future Lifetime
Probability Functions
Actuarial Notations
# $ = Probability that survives years
= Pr $ >
= $
# $ = Probability that dies within years
=
Last updated May 25, 2016.
1. [First Pass] Which of the following are true?
1.
t|u qx
= t px
2.
t|u qx
=
3.
t|u qx
= t px
u qx+t
lx+t+u lx+t
lx
t+u px
A. 1
B. 2
C. 3
D. 1, 2
E. 1, 3
2. [First Pass] Given that a life aged 50 will live to age 60, what is
B.1 Survival Distributions and Life Tables
B.1.1 Probability Functions
Future Lifetime of a newborn
Future Lifetime of pxq
Cumulative Distribution
Survival Distribution
Curtate Future Lifetime
Exercises
B.1.2 Actuarial Notation for Probabilities
B.1.3 Lif
Last updated May 25, 2016.
1. [First Pass] Which of the following are true?
1.
t|u qx
= t px
2.
t|u qx
=
3.
t|u qx
= t px
u qx+t
lx+t+u lx+t
lx
t+u px
A. 1
B. 2
Draw a picture for
C. 3
D. 1, 2
t|u qx
t px
u qx+t
t px
x
lx
(i)
t|u qx
= t px u qx+t
(ii)
t
Last updated May 25, 2016.
1. [First Pass] Which of the following are true?
1.
t|u qx
= t px
2.
t|u qx
=
3.
t|u qx
= t px
u qx+t
lx+t+u lx+t
lx
t+u px
A. 1
TIA MLC B1.3 Problems
B. 2
C. 3
1 of 11
D. 1, 2
E. 1, 3
2. [First Pass] Given that a life aged 50
B.1 Examples and Exercises
B.1.1 - Probability Functions
Slide 5/27:
Express the probability that a (30) dies between ages 40 and 45 using the cumulative distribution
function.
B.1 Examples and Exercises
1 of 7
Slide 7/27:
Express the probability that a (