Introduction
ELEMENTS OF GENERAL RISK THEORY
Vladimir Rykov
(E-mail: vladimir rykov@mail.ru)
Dept. of Appl. Math. and Comp. Modelling,
Russian State University of Oil&Gas
Introduction
OUTLINE
1. Introduction
2. Background
2.1
2.2
2.3
2.4
2.5
Risk notion
Risk measurement
Some classes of r.v.’s and operations with them
Some parametric families of risk distributions
Hierarchical systems and event tree
3. Engineering Risk Theory
3.1
3.2
3.3
3.4
3.5
3.6
Methodology of risk analysis
Structural reliability
Risk tree construction
Risk tree analysis
Qualitative analysis
4. Insurance Risk Theory (Ruin Problem)
Introduction
Introduction
During last years the term “RISK” became very popular in different
areas:
mathematics,
I engineering,
I economics,
I environment,
I management,
I biology & medicine, etc.
and different authors use this term in different senses.
I
Below we consider some definitions of risk in different sources.
Introduction
Dictionaries.
I
I
[Webster’s new universal unabridged dictionary (1996)]
“Risk is chance of losses, probability of losses”
[Ushakov commenting dictionary (1935)]
“Risk is possible dangerous, possible loss (damage) or non-success
in business”.
Introduction
International documents.
I
I
[Basic safety principles for nuclear power plants. 75-INSAG-3 (1999)]
“Risk R(t) for time t, connected with some event, is defined as a
product of the probability of this event by non-wished
consequence D of this event: R(t) = P(t)D”
[Functional safety of electrical/ekectronic/programmable
electronic-related systems (ISO/IEC Guide 51: 1990)]
“Risk is combination of the probability of occurrence of harm and
the severity of that harm”.
Introduction
National laws.
I
I
I
[Russian Federation law about technical regulation (2002)]
“Risk is a probability of harm causing for life or healthy of citizens,
property of physical or juristical persons, states or municipal
property, environmental, life or healthy of animals and plants with
taking into account the size of this harm”
[Russian Federation Federal Law No.7-FL.]
“Ecological risk is a probability of events occurrence, that has some
bad consequences for environment and caused by negative influence
of economical or other activity, extraordinary situations of natural
and technogeneous character.”
[Law of Ukraine (2005)]
“Risk is a possibility of occurrence and a probable scale of
consequences from negative phenomenon during some time
period”.
Introduction
Normative documents.
I
I
I
[Russian State Standard. Risk Management. Terms and Definitions
(2005)]
“Risk is a combination of probability of event and its
consequences”.
[Russian State Standard. Safety aspects. Rules for including in
standards (2002)]
“Risk is a combination of harm probability and its heaviness”.
[An American National Standard (ANSI/IEEE, 1987). IEEE Guide
for General Principles of Reliability Analysis of Nuclear Power
Generation Station Safety Systems]
“Risk is a measure of the probability and severity of undesired
effects. Often taken as the simple product of probability and
consequence”.
Introduction
Authors.
I
I
[Alpeev A. (2005)]
“Risk is an evaluation of expected damage from possible event in
given conditions”.
[Efimov S.L. (1996)]
“Risk is
1) a danger of a negative event”...”... Risk is characterized by a
collection of circumstances in their unity and interaction”,
2) an object of insurance”,
3) a kind of responsibility of the insurer”.
Introduction
I
[Commercial insurance in gas industry. M.: “Gasprom” (1998)]
“A wide spectrum of ... circumstances, not dependent from will of
the owner, having a property of probability, and unexpectedness of
appearance, are called as risk of economic activity of the subject”.
Introduction
I
I
[E.J. Henley, H.Kumamoto (1992)]
use a dictionary definition of risk: “Risk is the possibility of loss or
injury to people and property”
[J. Grandel (1991)] In the book under title “Aspects of risk theory”
there is no definition of risk at all, only descriptions of some risk
models.
Introduction
I
I
[Gordon B.G. (2003).]
“Risk R is the product of probability of event P by the value of its
consequences V : R = PV ”.
[Glenn Koller (2005)]
Risk is “. . . A pertinent event for which there is a textual
description. . . . Associated with each risk are typically at least two
parameters: probability of occurrence, consequence (impact) of
occurrence.”
Introduction
I
[N. Singpurvala (2006)]
Term risk arises in many contexts, but there is not its definition
and methods of measurement.
I
[E. Solojentsev (2006)]
Also term risk arises in many contexts, there are many parameters
of risk but its definition is also absent.
Introduction
Thus, the situation in this area now looks like a situation in reliability
theory more then half a century ago, when different specialists use the
term “reliability” in different senses and understood under reliability
different its characteristics.
Introduction
Historically among mathematicians and actuaries this term arisen in
framework of ruin models, Cramer (1930, 1955), Grandel (1991), a.o.
From another side, among engineers the notion risk is used in framework
of reliability theory and means the losses or damages for peoples,
environment and finance, Heynly&Kumamoto (1992).
There is one more conception of risk due to von Niemann and
Morgenstern (1953) which use utility theory for comparison of random
variables (r.v.). It is used for comparison of different financial and
business decisions utilities.
Introduction
Therefore, now there exist at least three different risk theories:
I Insurance risk theory, which deals with the ruin problems;
Engineering risk theory, which deals with investigation of the
consequences (damages), generated with failures of complex
non-reliable systems;
I Business risk theory, which considers the problems of r.v.
comparison in framework of utility theory.
In this situation the strong and common for different cases notion of risk,
that could be used in inter-disciplinary connections, and the methods for
the risk measurement and assessment are needed.
I
Introduction
The analysis of existing notion risk definitions shows that
I any authors
also as different national
and international documents
define this term quite differently, including into definition different
characteristics of risk.
I
I
This shows the necessity to propose the general definition of risk
understandable and useful for different kind of specialists.
Introduction
I. BACKGROUND
Chapter I. BACKGROUND
I.1. Risk notion
Different kinds of risks pursue both individuals during whole their life and
juridical subjects:
I industrial,
I agricultural plants,
I financial,
I insurance companies,
I societies,
I political communities,
I biological objects,
I ecological structures etc.
Introduction
Most of economical, technical, political social and others decisions
connected with “risk”, that is inevitably arise due to randomness and
uncertainty of factors influencing to the phenomenon to which the
decision is taken.
Habitual idea about “risk” means an occurrence some random “risk
events”, and its consequence such as financial, material or others loses.
To separate a common component and differences in different situations,
which in habitual life are joint as risks situations, to show the problems of
general risk theory, and to formulate the goal and mean notions of
general risk theory, consider several examples.
Introduction
1.1. Examples
1.1.1. Risk of the health lost
Crossing the street in the icy time one has a risk to fall and twist a leg or
to get another injury. Here risk arise as a (random) event, and its
consequence is the cost of treatment, or missing reward.
1.1.2. Risk of work ability lost
Some types of professions contains the risks of lost of able to work before
the retire time as a result of professional illness. Another types of
work-ability lost risks arise as a result of technological or social changing
in the community. It is obvious that the able-bodied lost also leads to the
financial and moral loses.
Introduction
1.1.3. Property missing risks
Driving one has a risk of accidence. Opening new business one has a risk
to loss the money, if it is not going well. Some person or a factory risk
with its wealthy in the case of fire, storm, earthquake etc. These
phenomenons are also distributed in time and also brings some loses.
1.1.4. Social risks
Because there are something to lose for the owners, they need to spend
some money for the life, health, property etc. defence.
Nevertheless, the adequate social organization of community might be
more direct and reliable way for defence of the interests of different layers
of society.
The problem what this organization should be is not the question for this
course. Another example of social risks are medical risks.
Introduction
1.1.5. Medical risks
For surgical operation we are risking with some disability and even the
life in more degree with less experienced surgeon. Nevertheless, as a
Russian writer and physician V.Veresaev noted, that because the
experience comes only with practice, thus if nobody will use the young
surgeons, the whole community risks to lost the professional surgeons.
This example shows a contradiction between an individual and social
risks.
Introduction
1.1.6. Financial and business risks
The play at a stock-market also as any financial operation associated
with risks.
1.1.7. Natural, ecological and technogeneous risks
The nature itself from one side and the humans’ activity from the other
side are sources of risks. The avoidances on large chemical enterprizes,
breaks in oil- and gas- pipelines etc. represent significant hazard and lead
to high damage for population and environment. Since these risks are
connected with reliability of appropriate equipment, their study directed
to excuse the payment for the providing and support of the necessary
level of reliability equipment.
Introduction
1.1.8. Insurance of risks and risk of insurance
Using the idea of risks accumulation (collectivization) the insurance
serves to stabilization of economics by means of smoothing of the risks
consequences. From another side an insurance itself is a source of risks,
which is connected with claims payment. More about insurance risks will
be done later.
The most of examples above deals with separate (individual) risks. But
it is necessary to take into account that in many of the above examples
(especially in insurance problems) risk situations can be repeated in time
that leads to the necessity to study so called collective risks.
Introduction
The variety of these examples demonstrates the difficulties of general risk
theory construction.
To do that one should separate the common part of all considered (and
many others) situations, to formulate the notion of risk and to propose
the measure and the tools for measuring and comparison of risks.
Introduction
Linguistic analysis and definition of risk notions
The linguistic analysis of the examples above shows the in any risk
situation the expert deals with
NEGATIVE
CHANGES
of
CONDITION
of
an
OBJECT
under
inner
or
exter
nal
This analysis justifies the following substantial definition
AFFECTS
or
CIRCUMSTANCES
Introduction
Definition
Definition
Risk is an event or sequence of events, which leads to negative changes
in an object states under inner or external affects or circumstances.
These events occurs in time and accompany with different damages.
This definition is not enough strong, but allows to propose some general
mathematical approach to model any risk situation.
Introduction
1.2. Notion of risk
In all of these examples risk is connected with occurrence some
I uncertain event A, which is called risk event
I
from the family F of events, describing considered risk situation,
and it is characterized by two values:
the time up to the event occurs (T ), and
I
the size of damage (X ).
I
Introduction
There are different types of uncertainties in the world. Accordingly to
Kolmogorov approach probabilistic type of uncertainty is characterized by
two main restrictions:
the possibility (at least in principle) infinitely many times observe
the phenomenon considered,
I under homogeneous conditions.
In this case statistics is the real tool for probability estimation.
I
Introduction
Therefore, from mathematical points of view the situation could be
considered with different methods:
I
I
I
I
probabilistic,
subjective (by methods of subjective probabilities),
expert (by utility theory methods),
by the fuzzy sets technique, etc.
Introduction
In the following we limited ourselves mostly by probabilistic aspects and
methods of risks study.
From this point of view the risk should be described as a probabilistic
space (Ω, F, P) on which a two-dimensional r.v. (T , X ), or a sequence of
two-dimensional r.v.’s (Sn , Xn , n = 0, 1, . . . ) are determined.
Definition
A risk is a two-component r.v. (T , X ), or a sequence of two-components
r.v.’s (a marked point process),
(Sn , Xn , n = 0, 1, . . . ),
defined under some probabilistic space (Ω, F, P).
Remark. The second component X might be a multi-dimensional one or
even functional. Therefore, the term random element would be more
appropriate, but we will use the first more usual and simple.
Introduction
The given definition seems a very simple, however some peculiarities of
risk analysis should be taken into account. These peculiarities are:
I the uniqueness of any risk situation;
the long chain of causes, that is necessary to be taken into account
in risk situation analysis;
I absence of needed statistical data.
All these circumstances lead to the necessity to construct for any risk
model its own probability space, which is usually realized in terms of the
risk tree construction, that will be considered later.
I
Introduction
1.3. Classification of risk models
The above examples give the possibility to classify risks accordingly to
their generation.
But this approach does not help for the general theory development,
since it distracts from general notions and methods of risk analysis.
Therefore in further we focus on general problems of risk modelling and
analysis.
The main problem of risk analysis consists in risk events uncertainties.
There are many types of uncertainties in the world, and the problem of
investigation and measurement of uncertainty is studied by many
scientists (see Bibliographical notes). We restricted ourselves in this
lectures mostly by the probabilistic one.
Introduction
There are two possible approach for random phenomenons modelling:
I theoretical (analytical);
I statistical.
The first one is certainly good, but it has limited applications, because it
is based on the delicate analysis of the phenomenon considered, that not
always possible.
The second one is theoretically universal, nevertheless not always
practically applicable because of necessity collection and elaborating
enough large information, which not always accessible because of its cost
and many of risk events are very rare.
Therefore in risk theory we need also in
I subjective (on the base of subjunctive probabilities and expert
analysis).
This approach has an evident defect, that is its subjectivity.
Introduction
Risk models traditionally are divided into:
I individual risks models and
I collective risks models.
An individual risk is risk, connected with one-point risk event though
possibly from wide family of events, while
A collective risk model deal with sequence of risks events, occurred in
time jointly with its damages. These phenomenons are studied in the
framework of risk processes.
Introduction
We will also divide individual risks into:
I simple risks and
I compound risks.
A simple risk is characterized by the possibility to evaluate or to
estimate its probabilistic characteristics directly.
A compound risk is characterized with many different events and leads
to numbers consequences. In this case it is natural to represent it as a
chain of components (simple risks), each of which could be described in
the framework of simple individual risks.
Introduction
In mathematical terminology
I
An individual risk is probabilistic model (Ω, F, P), on which
two-component r.v. (T , X ) is defined,
I
I
I
the first component T represents the time up to risk event A
occurrence beginning from some fixed epoch,
the second one X denotes a damage from this risk event.
A collective risk is probabilistic model (Ω, F, P), on which it is
defined a sequence of two-component r.v.’s,
{(Sn , Xn ) : n = 0, 1, . . . }
I
I
the first components Sn represent the time up to n-th risk event An
occurrence beginning from some fixed epoch,
the second ones Xn denote damages from these risk events.
Therefore construction and study of the probabilistic risk model for
different risk situations compose one of aspects of the mathematical risk
theory.
Introduction
The time T of risk event occurrence should be measured from some
natural “initial” epoch t0 of the process beginning.
In reliability theory this epoch is the equipment beginning
exploitation epoch.
I In the life insurance models appropriate epoch is the birth epoch.
I Nevertheless, in many practical situations (for example in some
catastrophic events in nature: earthquakes, tsunami, etc.) it is
impossible to fix some specific beginning epochs.
Note also that if a risk situation is studied in a fixed time interval, then
the risk event can not occurs during this interval at all.
I
Introduction
Therefore, three type of risk models should be considered.
I
I
I
Short-time model, where the probability of risk event occurrence is
mach more smaller then one
Middle-time model, where the probability of risk event occurrence
is smaller then one.
Long-time model, where the probability of risk event occurrence
equals to one.
Introduction
Risks Classification
RISKS CLASSIFICATION
-
NATURE
METHODS
engineering,
environmental,
production,
financial,
social,
insurance,
others.
-
stochastic,
subjective,
expert,
fuzzy.
TIME
- short,
- middle,
- long.
MODELS
- individual,
- compound,
- collective.
Introduction
1.4. Bibliographical notes.
The problem of investigation and measurement of uncertainty is studied
by many scientists beginning from Kardano (1501-1575), including
D.Bernoulli (1700-1782), De Moivre, De Finetty, Fermat, Huygens,
Laplace, Pascal, Poisson and others. Kolmogorov (1903-1987) specifies
probabilistic uncertainty that is characterized by two maim conditions:
I possibility to observe a phenomenon (in principle) infinity many
times,
I in homogeneous conditions.
In this case the problem of uncertainty is described in terms of
probability space (Ω, F, P) and mathematical statistics is a tool for the
probability measure estimation. We will refer to this type of uncertainty
to as randomness while safe the term uncertainty for all other its types.
Introduction
Historically the notion risk among of engineers arisen in framework of
reliability theory and means the losses or damages for peoples,
environment and finance. The methodology of engineering risk analysis
firstly was proposed in Bell Laboratory (for bibliography see Henley and
Kumamoto, 1992).
Insurance risks are studied mostly in terms of ruin problem. The ruin
problem also has a long history. The first works in this direction appears
due to Lundberg (1903) and in the framework of mathematical insurance
models belongs to H.Cramer, 1930. For some other details and
approaches see also H.Cramer, 1955, S.Andersen, 1957, W.Feller, 1966,
S.Asmusssen, 1996, Jan Grandell, 1991, and others.
The conception of risks comparison in the framework of utility theory
belongs to J. von Niemann and O. Morgenstern, 1953. The new results
and the bibliography in this direction one can find in the review of
V.Rotar and V.Bening, 1994.
Introduction
I.2. Risk measurement
In this section the main risk measure and some special its characteristics
will be proposed.
Introduction
2.1. Risk distribution
Above an individual risk was defined as a two-component r.v., both of
which without of lost generality can be supposed to be positive.
Therefore, it is determined by its cumulative distribution function
(c.d.f.)
F (t, x) = P{T ≤ t, X ≤ x}.
(1)
Remark. The damage itself is a multi-dimensional value, and moreover,
for some applications it can be functional. Therefore, it can be considered
as an element in some complex (may be functional) space. Nevertheless,
further we will limited ourself mainly with positive r.v. for damage.
Introduction
In most practical situations the information about joint distribution of
time and damage of risk event is not accessible, and we need to limited
ourselves with only marginal c.d.f. of times
FT (t) = P{T ≤ t} = F (t, ∞),
(2)
FX (x) = P{X ≤ x} = F (∞, x).
(3)
and damage size
Later in § 5 some parametric families of distributions for times and
damages description will be done.
Introduction
If risk is considered at fixed time interval, then instead of risk event
occurrence time T it is more convenient to consider the probability P(A)
of the risk event A and a conditional c.d.f. of damage given A,
G (x; A) = P{X ≤ x|A}
with
G (0; A) = P{X = 0| A} = 0.
In this case unconditional damage size has a form with a jump at zero,
because there is no damages if risk event does not occurs,
FX (x) = 1 − P(A) + P(A)G (x; A)).
Introduction
In general, it is more convenient to measure risk by the distribution of
risk event time occurrence FT (t) = F (t) and conditional risk damage
distribution given T
G (x; t) = P{X ≤ x|T = t}.
(4)
Therefore, their joint distribution is
Z t
F (x, t) =
G (x; u)dFT (u).
(5)
0
In the simplest case it is supposed that time and damage are independent.
G (x; u) = G (x) = FX (x) and F (x, t) = G (x)F (t).
Introduction
In many practical cases this assumption is quite admissible with only the
remark that the future damages at given time should be discounted with
some discount rate s and therefore the future damage is measured with
its present value, given with
X̂ = e −sT X .
Really, to cover the damage of the size X after the time T it is enough
to put in the bank sum X̂ under s%. Therefore the c.g.f of damage
present value is
Z ∞
F̂X (x) = P{X̂ ≤ x} =
P{Xe −st ≤ x} dFT (t) =
0
Z ∞
st
=
FX (xe ) dFT (t).
(6)
0
Introduction
Everywhere later in this course it is supposed that r.v. T and X are
independent, and we will denote their c.d.f. as F (.), and G (.)
respectively.
In reliability (and engineering risk) theory the tail of the c.d.f. FT (t)
R(t) = 1 − FT (t) = P{T > t}
(7)
is usually called by reliability function (in demography and insurance it
is called with survival function). Following to these traditions it will be
called here by risk function.
Introduction
For continuous distributions these functions plot at the figure below
1
0.9
F (t)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
R(t)
0.1
0
0
1
2
3
4
5
6
7
8
9
10
t
Figure: C.d.f. time to risk event and reliability function.
Introduction
For continuously observed r.v. it is more convenient to model their
distributions with the help of probability density function (p.d.f.)
f (t) = F 0 (t)
and
g (x) = G 0 (x).
(8)
Therefore, the c.d.f. can be represented in terms of p.d.f. in the form
Z x
Z x
F (x) =
f (u) du
and G (x) =
g (u) du.
0
0
In most practical cases the time and the damage are measured in discrete
units. In this case an appropriate model is discrete distribution
fk = P{T = k},
gk = P{X = k}
(9)
where k represents the number of time or damage units. Discrete
distributions are also used for modelling compound distributions in § 6.
Introduction
2.2. Risk intensity and hazard rates
Since the times to risks events are measured usually from some special
time (or event), it is very important to know conditional distribution of
the residual time to risk event occurrence, given time t after “natural”
beginning of the process.
This is the conditional probability that the risk event occurs in time
interval (t, t + x], given that it does not take place before the time t. (In
the formula below it should not be confused with two-dimensional
distribution (1)).
FT (x; t)
P{t < T < t + x}
=
P{T > t}
F (t + x) − F (t)
R(t) − R(t + x)
=
.
(10)
1 − F (t)
R(t)
= P{T ≤ t + x|T > t} =
=
Introduction
For the small values of x one can get:
F (x, t) =
f (t)
x = λ(t)x.
1 − F (t)
More strongly the function λ(t) is defined by the formula
1 F (t + ∆) − F (t)
f (t)
=
,
∆→0 ∆
1 − F (t)
1 − F (t)
λ(t) = lim
(11)
where function λ(t) represent conditional “instantaneous” probability
density of risk events occurs at time t after the natural beginning of the
process and it is called risk hazard rate function. The integral of it
Z t
f (u)
Λ(t) =
du = − ln(1 − F (t))
(12)
0 1 − F (u)
is called a hazard function.
Remark. In reliability theory the function λ(t) is known as hazard rate,
and in demography it is known as mortality rate.
Introduction
With risk hazard rate function one can evaluate the probability of risk
event occurrence during a small interval ∆t at the time t after its natural
beginning time with the area under the curve as it is shown in the
figure 2.2, where typical mortality rate function is presented. Enough
high value of this function at the very beginning shows the enfant
mortality level.
λ(t)
λ
t
t+∆
Figure: Hazard rate function.
t
Introduction
The equality (12) allows to represent c.d.f. of time to risk event and the
risk function in terms of risk hazard function
Z t
1 − F (t) = R(t) = exp{−
λ(u) du} = exp{−Λ(t)}.
(13)
0
Analogously, for the probability of risk occurrence in the time interval
(t, t + x] one can find
 t+x

 Z

F (x; t) = P{T ≤ t + x| T > t} = exp −
λ(u) du .
(14)


t
Introduction
Remark. Jointly with the h.r.f. in time in different models it might be
useful to consider also the hazard rate in space. It might be called as
intensity (rate) of risk expansion. For example it is possible to suppose
that the forrest fire expansion has a downward risk rate expansion. This
means that the probability of localization and stopping of fire at the
beginning stage is enough high, and it decreases with its expansion up to
some limit, defined by another factors, after that it could increase again.
Introduction
2.3. Generating functions and their properties.
The generating functions is a very useful tools for different characteristics
of risk calculation.
Definition
Let G is a distribution of a non-negative r.v. X , then the function
Z∞
g̃ (s) = Ee
sX
Z
sx
=
e G (dx) =
0
∞
e sx g (x)dx
0
is called the moment generating function (m.g.f.) of r.v. X , and/or its
c.d.f. G (x).
Introduction
Note that up to the sign in power it coincides with the Laplace transform
of the p.d.f. or Laplace-Stiltjes transform of the c.p.f. of r.v. X .
Z∞
g̃L (s) = Ee
−sX
=
Z
e
0
−sx
G (dx) =
0
∞
e −sx g (x)dx = g̃ (−s).
Introduction
Definition
Let {pk , k = 0, 1 . . . } is a distribution of an integer-valued r.v. N, then
the function
∞
X
z k pk
p(z) = Ez N =
k=0
is called the probability generating function (p.g.f.) of r.v. N, and/or
its distribution {pk , k = 0, 1 . . . }.
Introduction
Remark 2.1. The coefficients of Taylor expansion of m.g.f. at point
s = 0 give the moments of r.v. X , and the coefficients of Taylor
expansion of p.g.f. at point z = 0 produce up to known multipliers the
distribution of r.v. N.
Remark 2.2. In spite of m.g.f. and p.g.f. are defined differently for
different r.v.’s, there exist a close connection between them, and they are
connected with one more transformation of r.v. distributions, namely
with characteristic functions. However, we will not touch of these
connections, since in further we will not use it. In the context for both of
these functions we will the term “generation functions” (g.f.).
Introduction
The application of these function is based on their properties, that
represented in the following theorems.
Theorem
[2.1.] A m.g.f. g̃ (s) of any non-negative r.v. X is unique determined in
all region Re s ≤ 0 of a complex variable s. A p.g.f. p(z) of any
integer-valued r.v. N is unique determined in all region |z| ≤ 1 of a
complex variable z. The distributions of appropriate r.v.’s are uniquely
reconstructed with their g.f.’s.
Introduction
Theorem
[2.2] G.f. of the sum of independent r.v.’s equals to the product of g.f.
of summands. In other words, g.f. of the convolution of distributions
equals to the product of their g.f.’s.
Proofs of these theorems contain in any textbook on probability theory,
and it is omitted here.
¥
Remark. These functions also can be considered as a functions of real
value. In this case they are connected with absolutely and completely
monotone functions.
Introduction
Definition
A function f (z) of real value z is called absolutely monotone at the
segment [a, b], if it is infinitely many times differentiable inside of this
segment and f (n) (z) ≥ 0 for a < s < b; a function f (s) of real value s is
called completely monotone at the segment [0, ∞], if it is infinitely
many times differentiable for s > 0 and (−1)n f (n)(s) ≥ 0 for s > 0.
Theorem
[2.3] P.g.f. is analytical inside 0 ≤ z ≤ 1 and all its derivatives are
positive, while LST of any non-negative r.v. is analytical for s ≥ 0 and all
its derivatives are sign-alternating. In other words p.g.f.’s are absolutely
and LST of any non-negative r.v. are complete monotone functions.
Proof can be find in Feller, 1966, and it is omitted here.
¥
Remark. As an exercise it is proposed to transform this theorem also for
m.g.f.
Introduction
Applications of the absolutely and completely monotone functions are
based on the following theorems.
Theorem
[2.4]
1. If f and g are complete monotone functions, then fg is also
complete monotone function.
2. If f is a complete monotone function, and g is a positive function
with a complete monotone derivative, then the function f (g ) is also
a complete monotone one.
Proof should be fulfilled as an exercise.
¥
Introduction
One more useful property of g.f.’s concerns to the calculation of sums of
random number of r.v.’s. It is based on the following theorem which is a
corollary from theorems 2 and 4.
Theorem
[2.5.] If p(z) is a p.g.f. of a integer-valued r.v N, and g̃ (s) is a m.g.f. of
i.i.d. r.v’s Xi i = 1, 2, . . . independent of N, then the m.g.f. of
compound r.v.
N
X
Xi
Y =
i=1
equals to
g̃Y (s) = p(g̃ (s)).
(15)
Introduction
Proof is obtained by the simple calculation with the help of the complete
probability formula,
g̃Y (s) = Me −sY =
∞
X
k=0
∞
X
£
pk M e −sY |N = k] =
pk g̃ k (s) = p(g̃ (s)).
¥
k=0
These properties are widely used for different characteristics of compound
distributions calculation, especially for calculation of the compound
distributions moments.
Introduction
2.4. Moments and other risk characteristics.
Not only functional characteristics of risk, but also some numerical,
especially mean value and variance for the time up to risk event, and
appropriate damages are interesting in practice.
Z∞
µT
=
ET =
Z∞
Z∞
t f (t) dt = (1 − F (t) dt = R(t) dt
0
σT2
=
0
0
Z∞
DT = E(T − µT )2 = (t − µT )2 f (t) dt.
(16)
0
Z∞
µX
=
EX =
Z∞
x g (t) dx = (1 − G (x)) dx
0
σX2
=
DX = E(X − µX )2 =
0
Z∞
(x − µX )2 g (x) dx.
0
(17)
Introduction
Remind that mean characterizes the “center of probabilistic weight” of a
r.v., while the variance characterizes of dispersion of appropriate r.v.
around its mean value.
Remark 2.1. Analogous expressions for discrete r.v. should be presented
in terms of appropriate sums.
Remark 2.2. In finance mathematics the variance is often used as risk
notion.
The applications of these methods for the risk moments calculation will
be considered in § 6.
Introduction
2.5. Additions
Examples.
1. Let {pk } be Poisson distribution with mean λ > 0,
pk =
λk −λ
e , k = 0, 1, 2, ...
k!
Acc. to def. its p.d.f. is
p(z) = e −λ(1−z) .
Introduction
2. Let {pk } be Poisson distribution with mean λ > 0. Calculate the
convolution
g = p (1) ∗ ... ∗ p (n)
Taking into account that {p (j) } equals to
pj (z) = e −λj (1−z)
and using the second part of the Th. 2 for g (z) one can find
g (z) =
n
Y
j=1
or
pj (z) =
k
Y
e −λj (1−z) ,
j=1
g (z) = e −(λ1 +...+λk )(1−z) .
Now from the first part of the Theorem it follows that g is the
Poisson distribution with mean λ1 + ... + λk .
Introduction
3. Using generating functions method find compound damage. If ϕ(s)
is a m.g.f. of damages Yi , then from part 2) of the theorem it
follows that
Z
∞
∞
X
X
pk [ϕ(s)]k .
pk e −sx g ∗k (x) dx =
ψ(s) =
k=0
k=0
Let
p(z) =
∞
X
pk z k
k=0
be the p.g.f. of the damages number N. Then
ψ(s) = p(ϕ(s)).
Introduction
Exercises.
1. Proof the first part of the Theorem 2.1.
2. Proof the Theorem 2.2.
3. Proof the Theorem 2.3.
4. Proof the Theorem 2.4.
5. Proof the Theorem 2.5.
Introduction
I.3. Some classes of r.v.’s and operations with them
In this section some special operations with non-negative r.v.’s, and some
classes of distributions such r.v.’s will be considered.
Introduction
3.1. Some operations with r.v. and their distributions
For risk analysis, especially for damage calculating some special
operations under r.v.’s and their distributions are needed. In this section
we consider some special techniques for this.
3.1.1. Shift.
Some times jointly with r.v. X it is necessary to consider also r.v.
Y = X + a, where a is some constant. Such an operation is called as a
shift, the c.d.f. of shifted r.v. equals
FY (x) = P{X + a ≤ x} = FX (x − a),
and plotted at the picture below
Introduction
FHxL
Λ=5, a=0.4
1
0.8
0.6
FX HxL=1-e-Λ x
FY HxL=1-e-Λ Hx-aL
0.4
0.2
x
0.25
0.5
0.75
1
1.25
1.5
1.75
Figure: A shifted c.d.f.
Introduction
Example
If somebody want to buy a house with help of some middleman with
fixed payment a for help, his real price for the house will be distributed as
a shifted by a price the house, Y = X + a.
Introduction
3.1.2. Scaling.
For the unit of measuring changing and/or for reducing different r.v. to
the same unit of measure the operation of scaling is used. This
operation consists in multiplying of a r.v. by some constant a. The c.d.f.
of scaled r.v Y = aX equals
³x ´
FY (x) = P{aX ≤ x} = FX
.
a
and it is plotted at the picture below
Introduction
FHxL
Λ=5, a1 =0.4, a2 =1.4
1
0.8
FX HxL=1-e-Λ x
x
x
FX H €€€€€€€€ L=1-e-Λ €€€€a1€€
a1
x
x
FX H €€€€€€€€ L=1-e-Λ €€€€a2€€
a2
0.6
0.4
0.2
x
0.2
0.4
0.6
0.8
Figure: A scaled c.d.f.
Introduction
Example
If some damages X should be recalculated from euros to USD it should
be multiplied by the current cost of euros in USD, for example a = 1.5.
Therefore Y = 1.5X .
Another situation of this type arise when a middleman want to take for
his work a percents from the prise of the house from the previous
example. In this case the real price of the house is a scaled by 1 + 0.01a
initial price of the house.
Introduction
3.1.3. Truncated distribution.
To truncated distribution one refers to as a conditional distribution of a
r.v. under condition that it takes its values only in some given subset of
its previous values. This operation is used both in different theoretical
calculation and in some applications. A distribution F (.) truncated, for
example, to the segment [ab], is
F[ab] (x) = P{X ≤ x|a ≤ x ≤ b} =
FX (x) − F (a)
1{a≤x≤b} .
FX (b) − FX (a)
Introduction
An appropriate graph at the figure below is shown
FHxL
1
Λ=5, a=0.4, b=1.4
0.8
0.6
FX HxL=1-e-Λ x
0.4
e-Λ a - e-Λ x
€€€€€€€€€€€€€€€€€€
F@a,bD HxL= €€€€€€€€€€€€€€€€
e-Λ a - e-Λ b
0.2
x
0.2
0.4
0.6
0.8
1
1.2
1.4
Figure: A truncated c.d.f.
Introduction
Example
In some insurance agreements the damage X that is smaller than some
fixed constant a does not covered. This constant is known as a
“franchise”. In this case insurance claim Y equals to the damage
truncated from the below Y = max{X , a}.
Another type of this situation arise, for example, in reinsurance, when the
main insurer take for himself risks with the damage not grater than a,
and a reinsurer cover the claims which are grater than a. Then the main
insure claim is the truncated from the above by a damage, and the
reinsurer’s claim is truncated from the below by a damage. Their c.d.f.
at the picture ? are shown
Introduction
3.1.6. Mixture.
At last consider the situation, when phenomenon of “risk” describes as
some family of risk events, from which only one is realized in concrete
situation.
This is usual situation in engineering risk models, where several different
risk events can occurs.
Suppose that this family contains finite or denumerable set of disjoint
events (any denumerable family can be reduced to disjoint one),
F = {Ak : k = 1, 2, . . . },
each of which leads to its own damage Xk with c.d.f. Gk (x).
Introduction
Denoting with 1{Ak } the indicator functions of these events and with
pk = P(Ak ) their probabilities, the resulting damage Y can be
represented as
∞
X
Y =
Xk 1Ak .
k=1
The c.d.f. G (x) of such damage is
G (x) = P{Y ≤ x} =
∞
X
k=1
and it is called by mixture of c.d.f.’s Gk (.).
pk Gk (x)
(18)
Introduction
Example
An insurance company propose different covering in different accidents of
life insurance:
I illness,
I dentist’s accidents,
surgical operations,
die, etc.
If covering in these cases are r.v. Xk , then the full covering is their
mixture with distribution
I
I
G (x) = P{X ≤ x} =
n
X
P(Ak )P{Xk ≤ x}.
k=1
An example of mixture of two exponential distributions at the figure
below is presented
Introduction
FHxL
Λ1 =1, Λ2 =3, p1 =0.3, p2 =0.7
1
0.8
0.6
G1 HxL=1-e-Λ1 x
G2 HxL=1-e-Λ2 x
GHxL=p1 G1 HxL+p2 G2 HxL
0.4
0.2
x
1
2
3
4
Figure: A mixture of two exponential c.d.f.
Introduction
3.1.7. Integrated tail distribution.
The distribution
1
F (x) =
µF
Zx
s
(1 − F (u)) du,
0
R∞
where µF = 0 (1 − F (u)) du is the mean of distribution F is known as
a integrated tail distribution for given distribution F .
It is clear, that this distribution exists only for distributions with finite
mean values.
This distribution arise, for example, in reliability theory for stationary
distributions of the “age” and the “residual life time” of elements in
repairable systems, which is modelled as renewal processes.
Remark. For exponential distribution the integrated tail distribution is
the same.
Introduction
Example
Consider accelerated reliability trails of some item with accelerated life
time T distributed with p.d.f. F (t). Being failed each item
instantaneously replaced by the new one. Then at some fixed enough
long after the very beginning time t0 the residual time up to the fail has
an integrated tail distribution
Below the graph of an integrated tail distribution is done.
Introduction
FHxL
Α=0.3, Β=3
1
0.8
x Α
FHxL=1-e-I €€Β€€ M
0.6
Α
G H €€Α1€€ L - G I €€Α1€€ , I €€xΒ€€ M M
Fs HxL= €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€
€€€€€€€€€€€€€€€€
€€€€€€€€€€€€€€ €
G H €€Α1€€ L
0.4
0.2
x
200
400
600
800
1000
Figure: An integrated tail c.d.f.
Introduction
3.1.4. Summation.
If X1 , ..., Xk are independent r.v.’s with c.d.f.’s Gi (x) and p.d.f.’s gi (x),
then the c.d.f. G (∗k) (x) of their sum
Y = X1 + ... + Xk
is given for all i = 1, k with the recursive formula
Zx
G
(∗0)
(x) = 1{x≥0} ,
G
(∗i)
G ∗(i−1) (x − u)gi (u)du,
(x) =
0
which is known as a convolution of Gi (x) and is denoted by
G (∗k) (x) = G1 ∗ G2 ∗ · · · ∗ Gk (x).
(19)
Introduction
In the case of i.i.d. r.v. the last formula takes more simple form
Zx
G
(∗0)
(x) = 1{x≥0} ,
G
(∗i)
G ∗(i−1) (x − u)g (u)du.
(x) =
(20)
0
For discrete i.i.d. r.v. Xi with the common distribution
gj = P{Xi = j}
j = 0, 1, . . .
the formula (20) is changed to the following one
(∗0)
gj
= 1{j=0} ,
(∗k)
gj
=
j
X
i=0
(∗(k−1))
gj−i
gi .
(21)
Introduction
Example
If some insurance company has n agreements for a fixed period of time,
and the claim for i-th agreement is Xi . Then the full claim of the
company during this period equals to Y = X1 + · · · + Xn .
Introduction
3.1.5. Compound distributions.
The sum Y of random number N of random components Xi (i = 1, N) is
known as compound r.v. and its distributions as compound
distribution. Put
(
0,
for
N=0,
Y =
X1 + X2 + ... + Xk , for
N=k=1,2,3....
Denote the distributions of r.v.’ N and Xk k = 1, N by
pk = P{N = k},
k = 0, 1, 2, ...
and
Gk (x) = P{Xk ≤ x}.
Introduction
Then taking into account that the c.d.f. of the sum X1 + ... + Xk of fixed
number of independent r.v.’s is G (∗k) (x) one can get
P{Y ≤ x} = p0 1{x≥0} +
∞
X
pk G (∗k) (x).
(22)
k=1
Notice that the index k here is not any more a r.v. but an integer, and
the c.d.f. G (∗k) (x) is determined with the convolution of c.d.f. of r.v.
X1 , ..., Xk .
Introduction
Example
If some tourists firm organize a bus tour, and a risk event results with
random number N claims, each of which is a r.v. Xk , k = 1, N. Then
the whole payment of the insurance company is a sum
Y = X1 + · · · + XN .
We turn to this situation later, when compound damage distributions will
be studied.
For calculation of distributions of sum and compound damages it is
necessary to use the convolution formula, which is not enough convenient
for calculation. In these cases a convenient tools are characteristic or
generating functions.
These functions are used for modelling, analysis, and different
characteristic calculation both time to risk event occurrence and values of
damages.
Introduction
3.4. Additions
Exercises.
Propose your own examples for
I shifted,
I scaling,
I truncated,
I mixture
distributions
Introduction
II.4. Some parametric families of risk distributions
One of the main problems of the risk theory is the modelling of
I times to risk event arising and
I damages from the event,
which is based on the characterization properties of distributions.
The nature of arising and development of risk situations is usually enough
complex. Thus, it is expedient that jointly with simple “unite” risks it is
also need to consider models of compound risks for enough complex
situations.
Because the same parametric families of distributions can be used for
modelling of both time to risk event arising and the damage from it we
consider some parametric families and classes of non-negative r.v.
distributions, that can be used for both of them. Some special
characterization properties of these distributions also will be considered.
We start with some discrete distributions.
Introduction
4.1. Some discrete parametric families of distributions
The most of observations are fixed in discrete of time or damage units,
therefore continuous distributions are only useful models to describe real
situations. Thus, in applications for describing both the time up to risk
event and the value of damage the different types of discrete distributions
are used. Moreover, discrete distributions are used in models of
compound damages for number of risks description.
In table 4.1 at the end of this section some discrete distributions jointly
with their main characteristics, which are often used in risk practice, are
given.
Consider some of them jointly with their g.f., means and variances.
Introduction
4.1.1. Degenerated distribution
The degenerated distribution is used for description of a non random
r.v. Its c.d.f. has a stepwise graph with a jump of unit size at point b,
G (x)
6
G (x) = 1{x≥b}
(23)
1
0
b
x
Fig. 2.4.1.1. Degenerated distribution
The m.g.f. mean value and variance for this distribution are
g (z) = Ez X = z b ,
EX = b,
DX = 0.
Introduction
Mixture of such types distributions allows to construct two-point
distribution and any discrete distribution as well, that often used for risk
characteristics modelling. The degenerated distribution and its mixture
can be used for claims modelling in different insurance models.
Example
4.1.1 Consider some contract of a short-time life insurance, that provides
two risk cases:
I a natural death – event A1 , and
I the death resulted by some accident – event A2 .
Suppose that the probabilities of these events and the claims provided by
insurance company are:
P(Ai ) = pi , bi , i = 1, 2.
Then the claim p.d.f. will be the two point step-wise function, shown at
the figure 2.4.1.1.1b.
Introduction
G (x)
6
1
p1
0
b1
b2
-
x
Fig.2.4.1.1b. Mixture of degenerated distribution
Introduction
4.1.2. Uniform distribution
The uniform distribution is characterized with fixed number, say, n of
values of r.v., each of which has the same probability,
P{X = i} =
1
n
1, n.
(24)
The m.g.f., the mean value and the variance of this distribution are
Ez X =
z 1 − zn
,
n 1−z
EX =
1
(n + 1),
2
DX ≈
4(n3 − 1) − 3(n + 1)2
.
12
The graph of c.d.f. has a step-wise form that is presented below at the
figure 2.4.1.2.
Introduction
1
F (x )
5/6
2/3
1/2
1/3
1/6
x
0
-1
0
1
2
3
4
5
6
7
Figure: Fig.2.4.1.2. The graph of uniform distribution.
8
Introduction
The uniform distribution can be used for damage distribution modelling
in the cases, when there is not enough information.
Example (4.1.2)
Suppose that the repair of a car after an accident costed
bi = $100i, (i = 1, 100), and all these payments are equally-probable.
This damage is described with the uniform distribution.
Introduction
4.1.3. Bernoulli distribution
This distribution is used for description of two-valued r.v. By choose the
scale and shift it is possible to reduce these values to zero and one,
pk = p k (1 − p)1−k ,
k = 0, 1.
Its g.f. equals
p(z) = 1 − p(1 − z),
and mean and variance are
µ = EX = p,
σ 2 = DX = p(1 − p).
The sequence of i.i.d. Bernoulli distributed r.v.’s constitute so called
Bernoulli scheme.
The graph of this distribution is presented at the figure 2.4.1.3.
(25)
Introduction
G (x)
6
1
1−p
- x
0
1
Fig.2.4.1.3. Bernoulli distribution
Introduction
Example (4.1.3)
Numerous observations during long time over all world show that for
among any 1000 birth approximately 515 are boys and 485 are girls.
Therefore, if one introduce a r.v., that takes value 1 for boy birth, and
value 0 for girl birth, it will have a Bernoulli distribution with p = 0.515.
Remark. Using shift and scaling operation any two point distribution also
could be modelled with Bernoulli distribution. If B is Bernoulli r.v., then
Y = b1 + (b2 − b1 )B,
where
is two point distributed r.v. see fig. 2.4.1.1b.
b1 < b2
Introduction
G (x)
6
1
p1
0
b1
b2
-
x
Fig.2.4.1.1b. Mixture of degenerated distribution
Introduction
4.1.4. Geometric distribution
This distribution is defined with the formula
pk = (1 − p)p k ,
Its p.g.f. is
p(z) =
k = 0, 1, 2, . . .
(26)
1−p
1 − pz
and two first moments are
µ = EX =
p
,
1−p
σ 2 = DX =
p
.
(1 − p)2
This distribution describes the moment of the “first success” in Bernoulli
scheme.
The graph of this distribution at the figure 2.4.1.4 is shown.
Introduction
g k (2/3)
0.3
0.2
0.1
k
0
5
10
15
20
Figure: Geometric distribution (p = 23 ).
25
30
Introduction
Among discrete distributions this is the unique which possesses of the
luck of memory property.
Lemma (4.1.1)
The luck of memory property
P{T ≥ k + l|T > l} = P{T ≥ k} = p k
(27)
is a characterization property of geometric distribution among discrete
distributions.
Prove this lemma as an exercise.
¥
Introduction
Example (4.1.4(a))
If some couple plan to have just one boy in their family, the number of
children will be distributed geometrically with parameter p = 0.515.
Example (4.1.4(b))
If in some telecommunication system a signal is transmitted with
probability of error equals to q = 1 − p, them the length of right
transmitted symbols will be geometrically distributed with parameter p
Introduction
4.1.5. Binomial distribution
The binomial distribution
µ ¶
n k
pk =
p (1 − p)n−k ,
k
k = 0, 1, . . . , n,
depends on two parameters: natural n and real p. Its m.g.f. is
p(z) = (1 − p(1 − z))n ,
and two first moments are
µ = EX = np,
σ 2 = DX = np(1 − p).
This is the distribution of number of “successes” in Bernoulli scheme
with n trials.
(28)
Introduction
The graph of the distribution at the figure 5.1.5 it is presented.
0.16
b k (2/3)
0.12
0.08
0.04
k
0
0
5
10
15
20
25
30
Figure: The shape for binomial distribution (n = 30, p = 23 ).
Introduction
Example (4.1.5.)
Some previous geological investigations show the presence of oil in given
region with probability p. If it is planed to bore n oil holes, the number of
non-empty oil holes among them will be described by binomial
distribution with parameters n, p.
Introduction
4.1.6. Negative binomial distribution
The negative binomial distribution
µ
¶
α+k −1 k
pk =
p (1 − p)α ,
k
k = 0, 1, . . .
(29)
for integer α represents the distribution of the trails number k in
Bernoulli scheme up to α-th success, but it also can be spread out of any
real α. Its m.g.f. is
³ 1 − p ´α
p(z) =
1 − pz
and two first moments are
µ = MX =
αp
,
1−p
σ 2 = DX =
αp
,
(1 − p)2
The graph of this distribution at the figure 2.4.1.6.
Introduction
0.12
0.1
0.08
0.06
0.04
0.02
5
10
15
20
Figure: The Negative binomial distribution graph.
Introduction
Example (4.1.6.)
If an Oil-drilling company want to investigate some region up to α-th
non-empty hole, the whole number of holes needed satisfies to the
negative binomial distribution with parameters α, p, where p is the oil
presence probability in the considered region.
Introduction
4.1.7. Poisson distribution
This distribution is defined with the formula
pk =
λk −λ
e ,
k!
k = 0, 1, 2, . . . ,
where real λ > 0 is its parameter. Its m.g.f. is
p(z) = E −λ(1−z) ,
and two first moments are
µ = MX = λ,
σ 2 = DX = λ.
The following figure represents Poisson distribution.
(30)
Introduction
0.12
π k (15)
0.1
0.08
0.06
0.04
0.02
k
0
0
5
10
15
20
25
Figure: The Poisson distribution graph (λ = 15).
30
Introduction
The main property are stability against summing, which is represented by
the following lemma.
Lemma (4.1.2.)
The sum of two (and several) Poisson-distributed r.v.’s has the Poisson
distribution parameter of which equals to the sum of parameters of
summands.
Prove this lemma as an exercise.
¥
Due to this stability property, concerning in previous lemma, the Poisson
distribution is used in many practical cases.
Introduction
The above property admits the following generalization.
Theorem (4.1.1.)
The sum of “many small” i.i.d. discrete r.v. has approximately Poisson
distribution. Formally, if
P{Ni = 1} =
Then
lim P
n→∞

 X

1≤i≤n
λ
,
n
P{Ni > 1} =


Ni = k

=
λk −λ
e ,
k!
1
.
n2
k = 0, 1, 2, . . . .
Introduction
Example (4.1.7.)
Due to the stability property, contained in Lemma 2, the Poisson
distribution is very good apt for describing different kinds of event flows:
I number of particles in radio-nuclear disintegration,
I
I
number of calls to telephone station,
number of E-mails, arisen to some server, etc.
In table 5.1 the summary of the discrete distributions jointly with its
main characteristics is given.
Table 5.1. Some discrete distributions and their characteristics
Title
Distribution
Mean
value
Variance
Poisson
λk −λ
e , k = 0, 1, . . .
k!
λ
λ
λ>0
Geomrtric
k
(1 − p)p , k = 0, 1, . . .
p
1−p
p
(1 − p)2
0≤p≤1
Binomial
µ ¶
n k
p (1−p)n−k , k = 0, 1, . . . n, np
k
np(1 − p)
Table 5.1. Prolongation
Title
Negative
binomial
Distribution
µ
¶
α+k −1 k
p (1 − p)α ,
k
Mean
value
Variance
αp
(1−p)2
αp
1−p
k = 0, 1, . . . , 0 < p < 1, α > 0
p(p+ln(1−p))
− (1−p)
2 ln2 (1−p)
Logarithmic
−
pk
, k = 0, 1, . . .
k ln(1 − p)
0<p<1
−p
×
(1 − p)
1
ln(1 − p)
Introduction
4.2. Some parametric families of continuous distributions
Consider some standard parametric families of continuous non-negative
distributions, which is often used for the time to risk events and simple
damages modelling. In the tables 5.2, 5.3 at the end of this section some
standard families of such distributions are shown. Bellow some comments
for them are given.
Introduction
4.2.1. Uniform distribution
The uniform distribution is determined with its p.d.f.
g (x) =
1
1{a≤x≤b} ,
b−a
0 < a < b,
(31)
which has a rectangular form (see fig. 2.4.2.1), that gives it another
name, rectangular distribution. The constants a and b are parameters
of distribution. Its c.p.d. is


for t < a,
0,
t−a
F (t) = b−a
, for a ≤ t ≤ b,


1,
for t > b,
Introduction
The graph of this distribution is shown below.
1
F (x )
0.8
0.6
0.4
0.2
p (x )
x
0
-10
0
10
20
30
Figure: C.d.f. and p.d.f. for uniform distribution.
40
Introduction
The hazard rate function is


for t < a,
0,
1
λ(t) = b−t
,
for a ≤ t ≤ b,


does not defined, for t > b,
Mean value and variance for this distribution respectively are
MX =
a+b
,
2
DX =
(b − a)2
.
12
Characterization property of this distribution is equali-probable of any
values of r.v. at any equal subintervals inside of segment [a, b].
Introduction
This distribution can be used both for description of the times of risk
events occurrence and the damage value in situations, when only the
possible boundary values for time or/and value of damage interval are
known, and more detailed information about their distribution absents.
I As examples it can be used as property insurance model, when only
the boundaries for estimated property are known, but real value of
claim unknown.
I Another example could be the failure time of new equipment, when
there is no enough statistical data for it estimation.
Example (4.2.1.)
Under absence of more detailed information and statistical data it is
possible to suppose that the pipeline lifetime uniformly distributed on the
interval [0,30] (years).
Introduction
4.2.2. Exponential distribution
This distribution has a c.d.f.
F (t) = 1 − e −λt ,
(32)
where λ is a parameter. Reliability (survival) function is
R(t) = e −λt ,
(33)
The mean value and the variance equal
µT = MT = λ−1 ,
σT2 = DT = λ−2 .
The graph of exponential distribution is presented at the figure 2.4.2.2.
Introduction
1
0.8
F (x )
0.6
0.4
p (x )
0.2
x
0
0
2
4
6
Figure: Exponential distribution.
8
Introduction
The hazard rate function is constant and equals to the parameter of
distribution λ,
f (t)
λ(t) =
= λ.
(34)
R(t)
This property of the h.r.f. allows to consider this distribution as a model
of instantaneous failures. Moreover this property is a characteristic
property of the distribution
Lemma (4.2.1.)
Among continuous distributions the exponential distribution is only one
with constant h.r.f.
Proof. Really, the relation (34) shows that the h.r.f. is constant. The
inverse follows from the formula
Zt
λ(u)du = 1 − e −λt .
F (t) = −
0
¥
Introduction
Another characterization property of exponential distribution is its
“luck of memory” property, which contains in the following lemma.
Lemma (4.2.2.)
An exponential distribution is only one for which the following “luck of
memory” property holds
P{T ≤ t + x|T > t} = P{T ≤ x} = 1 − e −λx
(35)
Introduction
Proof. Really, using the formula for conditional probabilities one get
P{T ≤ t + x|T > x} =
=
P{T ≤ t + x, T > t}
P{t < T ≤ t + x}
=
=
P{T > t}
P{T > t}
e −λt − e −λ(t+x)
= 1 − e −λx = P{T ≤ x}.
e −λt
The inverse statement is obtained from
R(t + x) =
=
P{T > t + x} = P{T > t + x|T > x}P{T > x} =
P{T > t}P{T > x} = R(t)R(x).
¥
Introduction
Exponential distribution can be used as for description of the time to risk
events arise, as a model for damage. Due to its characterization
properties in reliability theory it is used for description of completely
random failures.
Example (4.2.2.)
Two girls talk by phone. After one subject they began to discuss another
one and so on. In this case at any point of time the longevity of residual
talk does not depend on the time spent to it up to now. Therefore, the
whole longevity of talk can be described by exponential distribution.
Introduction
4.2.3. Shifted exponential distribution
The p.d.f. of this distribution is (see fig. 2.4.2.3)
g (x) = λe −λ(x−b) 1{x≥b} ,
b ≥ 0.
(36)
Its mean value and variance are
MX = b +
1
,
λ
DX =
1
.
λ2
This distribution can be used both for the time till risk events and for the
damages modelling, when appropriate values do not take values less than
some given constant. For example in insurance it is used for claims
modelling in the case when the claims less than given value do not
compensated (franchise).
Introduction
0.1
0.08
0.06
0.04
0.02
–10
0
10
20
30
40
50
x
Figure: The shifted exponential distribution.
60
Introduction
Example (4.2.3.)
Automobile traffic at some not very loaded road can be described with
Poisson flow, that means that the intervals between cars are
exponentially distributed. Nevertheless, the movement rules demand to
have some special intervals between cars. Therefore, the real distribution
of the intervals between cars are shifted exponential one.
Introduction
4.2.4. Gamma-distribution
The p.d.f. of this distribution is (see fig. 2.4.2.4.)
g (x) =
λα x α−1 −λx
e
1{x≥0} ,
Γ(α)
α > 0,
(37)
R∞
where Γ(α) = 0 x α−1 e −x dx is the Gamma-function, and λ is a scale,
while α is a shape parameters. For α = 1 it coincide with p.d.f. of
exponential distribution.
Survival (risk) function equals to
Z∞
R(t) = F̄ (t) =
λt
x α−1 −x
e dx
Γ(α)
Introduction
0.1
0.08
0.06
0.04
0.02
0
10
20
30
40
50
x
Figure: P.d.f. of Gamma-distribution.
60
Introduction
Appropriate c.d.f. is presented at the figure 17
1
0.8
0.6
0.4
0.2
0
10
20
30
40
x
Figure: Gamma distribution.
50
60
Introduction
The closed formula for the hazard rate function does not exists, but
mean value and variance are
µT = MT =
α
,
λ
σT2 = DT =
α
.
λ2
The Gamma distribution has the following stability property.
Lemma (4.2.3.)
If
αi , λ, then
PTi : i = 1, 2 have the Gamma distributions with parameters
P
i Xi has the Gamma distributions with parameters
i αi , λ.
When α → ∞ Gamma distribution is approximated with normal one.
Example (4.2.4.)
The Gamma distribution is used for modelling non-symmetric unimodal
distributions.
Introduction
4.2.5. Lognormal distribution
The survival function for this distribution is
µ
¶
µ
¶
ln t − µ
µ − ln t
=Φ
R(t) = 1 − Φ
σ
σ
with p.d.f.
g (x) =
1
√
xσ 2π
e−
(log x−µ)2
2σ 2
1{x≥0} ,
and mean and variance
µX = MX = e µ+
σ2
2
,
¡ 2
¢
2
σX2 = DX = e σ − 1 e 2µ+σ .
(38)
Introduction
Appropriate c.d.f. is presented at the figure 18
1
0.8
0.6
0.4
0.2
0
1
2
3
4
x
Figure: Lognormal distribution.
5
6
Introduction
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
x
Figure: p.d.f. of lognormal distribution.
6
Introduction
Example (4.2.5.)
In reliability theory this distribution is used for repair time modelling.
From the other side, this is one of so called heavy-tailed distributions, and
it is used for modelling “large” (catastrophic) damages of risk events.
Introduction
4.2.6. Pareto distribution
This distribution has a p.d.f
g (x) =
α ³ c ´α+1
1{x≥c}
c x
c >0
with expectation and variance
µX
=
σX2
=
αc
for α > 1,
α−1
αc 2
DX =
for α > 2.
(α − 1)(α − 2)
MX =
(39)
Introduction
0.8
0.6
0.4
0.2
0
2
4
6
8
x
Figure: p.d.f. of Pareto distribution.
10
Introduction
Example (5.2.6.)
Pareto distribution is also belongs to the class of heavy-tailed
distributions, and it is used for large harms of risk event modelling.
Introduction
4.2.7. Gnedenko-Weibull distribution
The survival function for this distribution is
α
R(t) = e −λt ,
(40)
with mean value and the variance
µT = MT =
σT2
= DT =
¡
¢
Γ 1 + α1
1
λα
,
¡
¢
¡
¢
Γ 1 + α2 − Γ2 1 + α1
.
2
λα
The exponential distribution being a special case of this distribution when
α = 1. If T is a GW distributed r.v., then the r.v.
X = λT α
has an exponential distribution.
Appropriate c.d.f is presented at the figure below
Introduction
1
0.8
0.6
0.4
0.2
0
1
2
3
4
5
x
Figure: The c.d.f. for Gnedenko-Weibull distribution.
Introduction
Its h.r.f. is
λ(t) = αλt α−1
and it is presented at the figure.
Figure: The h.r.f. for Gnedenko-Weibull distribution.
Introduction
The popularity of this distribution in Reliability Theory is explained by
the property of this distribution to be as a limiting distribution for
maximum and minimum if series of i.i.d. r.v.’s.
Really, because the failure time of a series system is the minimum of
failure times of its components, and the failure time of a system in
parallel is the maximum of failure times of its components, the
following theorem is very important. Denote by
X(n) = max{Xi : i = 1, n},
and by
X(1) = min{Xi : i = 1, n}.
Introduction
Because X(n) increases when n increases, it is natural to find some
sequences of constants an , bn such that the limiting distribution of r.v.’s
Yn = an X(n) + bn
to be proper distribution.
when
n→∞
Introduction
There are two types of limiting distributions for Yn dependently on the
behavior of tails distribution of r.v.’s Xi ,
1 − F (x) = P{X > x}.
Let
1 − G (x) = lim P{Yn > x}.
n→∞
Theorem (4.2.1.)
The following holds:
x
I If 1 − F (x) ≈ e −x , when x → ∞, then 1 − G (x) = e −e .
α
I If 1 − F (x) ≈ x −α , when x → ∞, then 1 − G (x) = e −x .
Introduction
Proof. of this theorem is outside of the framework of this course, and it
is omitted.
¥
The second case of this theorem leads to the GW distribution, while the
first one corresponds to the Gompertz distribution, which in survival
analysis is used.
Introduction
The GW distribution possesses the following stability property.
Theorem (4.2.2.)
If independent r.v. Xi , (i = 1, n) have the GW distribution with
parameters (λi , α). Then the r.v.
Y = min Xi
1≤i≤n
P
has GW distribution with parameters ( 1≤i≤n , α).
Proof. can be got by direct calculation.
¥
Introduction
Example (4.2.7.)
The life time of a system in series with independent identical elements,
life time of each of which has a GW distribution, also has a GW
distribution
α
RS (t) = R1 (t) · · · Rn (t) = exp {− (λt α + . . . λt α )} = e −nλt .
Introduction
4.2.8. Truncated normal distribution
The truncated normal distribution can be used for description of the
epochs for partially forecasting risks, especially in reliability theory for
time to gradual failures.
To explain this settings note that
I if an item failure results of some physical parameter a goes out of
limiting value a1 , and
I
I
I
this parameter changes accordingly to deterministic rule a = f (t, a0 ),
from an initial random state a0 normally distributed,
then the time up to the parameter a reaches to critical value a1 also
has a normal distribution.
Introduction
Really, under described conditions the failure time is a solution of
f (T , a0 ) = a1 .
Denoting by ϕ(a1 , a0 ) the inverse function to f (t, a0 ), one get
T = ϕ(a1 , a0 ).
Expanding the function ϕ(a1 , a0 ) in the Tailor series with respect to a0
around the point a = Ma0 up to second order members, one get
T = ϕ(a1 , a) + ϕ0a (a1 , a)(a − a0 ).
From here it follows that if parameter a0 has a normal distribution, then
the failure epoch T also must be normally distributed. However, because
the time to failure has only non negative values, the distribution should
be truncated at zero.
Introduction
The risk function for this distribution is
¡
¢
¡
¢
1 − Φ t−µ
Φ µ−t
σ ¢
σ¢
¡
¡
R(t) =
=
,
1 − Φ − σµ
Φ σµ
where Φ(x) =
√1
2π
Rx
t ≥ 0,
(41)
u2
e − 2 du is the standard normal c.p.d. When (as
−∞
usually holds) σ << µ, thus it is possible to use an approximation
µ
¶
µ−t
F̄ (t) = Φ
.
σ
Expressions for the mean value and the variance of this distribution are
enough complicated, but for the usual situation, when σ << µ they
coincide with the parameters of normal distribution:
µT = MT = µ
σT2 = DT = σ 2 .
Introduction
The c.d.f. of the distribution is presented at the figure below
1
0.8
0.6
0.4
0.2
0
0.5
1
1.5
2
2.5
x
Figure: The c.d.f. for truncated normal distribution.
Introduction
The h.r.f. is
(t−µ)2
2
1
e − 2σ
λ(t) = √
× ¡ µ−t ¢ ,
Φ σ
2πσ
and as it can be shown numerically has an asymptote y =
looks like as shown at the figure 2.4.2.8.
(t−µ)
σ
and
Figure: The hazard rate function for truncated normal law.
Due to the h.r.f. property this distribution for degradation modelling is
used.
Introduction
4.3. Almost-luck-of memory distributions
Some generalization of exponential family is so called family of Almost
Luck of Memory (ALM) distributions. This family is used for modelling
periodic phenomenons in life and nature. Some examples are:
(i) periodicity in sun activity;
(ii) periodicity of seasons in year, of ocean floods-ebbs, etc.;
(iii) many risk events, such as auto accidents, fires, zunamis, epidemics,
pollution, etc.
Introduction
We start with the defining the concept of lack-of-memory (LM)
property for random variables.
Definition
A non-degenerated at zero non-negative r.v. X possesses the
LM-property at point c > 0 iff
P{X ≥ c + x | X ≥ c} = P{X ≥ x}
for all x ≥ 0. Naturally, it make sense only if 0 < P{X ≥ c} < 1.
Introduction
Note also if this property holds for all possible values c of r.v. X , then
accordingly to the lemmas 4.1.1 and 4.2.1 this r.v. has or geometric, or
exponential distributions.
Lemma (4.3.1.)
If r.v. X possesses LM-property at the point c > 0, then this property
holds for all points of sequence
{am = mc}∞
m=0 .
¥
This means that the time axis can be divided into intervals
[0, c), [c, 2c), . . . , [mc, (m + 1)c), . . . ,
in which the r.v. is renewed.
Introduction
The characterization property of the ALM distributions is
Theorem (4.3.1.)
Let r.v. X possesses of ALM property for sequence {cm = mc}∞
m=0 . Then
(i) for continuous X its p.d.f fX (x), x ≥ 0 is
fX (x) = (1 − a)a[x/c] fY (x − [x/c]c),
(42)
where a = P{X ≥ c} and fY (.) is p.d.f. of some continuous r.v. Y at
finite interval [0, c);
(ii) for discrete r.v.X its distribution pX (x) is
pX (x) = (1 − a)a[x/c] pY (x − [x/c]c),
(43)
where the value a is the same as before, and pY (.) is distribution of some
discrete r.v. Y with finite number of values {0, 1, . . . , c − 1}.
¥
Introduction
Another description of the ALM family convenient for simulation is
Theorem (4.3.2.)
A r.v. X possesses the ALM property at point c and, therefore, for the
sequence {cm = mc}∞
m=0 if and only if it can be represented as a sum
X = Yc + cZ
(44)
of independent r.v.’s Yc and Z , such that Yc is distributed at finite
segment [0, c), and Z has a geometric distribution
pZ (z) = (1 − a)az ,
z = 0, 1, . . . .
¥
Most of these distributions can be used for modelling both the time till to
simple risk events and the damage them resulted.
Some parametric families of distributions jointly with their characteristics
are represented at the tables 5.2, 5.3 in the end of this section.
Introduction
4.4. Distributions with monotone hazard rate
Some families of distributions are studied based on the properties of their
hazard rate functions.
I
I
For example constant h.r.f. characterized exponential or geometric
distributions.
This allow to model time to risk event arising with appropriate
distributions if there exists some theoretical bases to suppose that
their h.r.f. is constant.
In other cases from some theoretical reasons or based on statistical
observations one can suppose some special shape (form) of h.r.f.
For example the long-term observations of the peoples’ mortality
show some concrete shape of the h.r.f. for this phenomenon, which
is shown at the figure 4.3.1.
Introduction
λ(t)
λ
t
t+∆
Figure: Mortality rate function.
t
Introduction
There exists an understandable treatment of this mortality rate behavior.
Analogous behavior of h.r.f. is observed in another phenomena, for
example for aging units in reliability theory, where the h.r.f. for aging
units increases with its age.
In most risk situations the risk event (or risk process) is a result of many
different reasons. For these cases the models of compound distributions
are more applicable for time till risk evens as well as for caused them
damage. This class of distributions in the next section will be considered.
Table 5.2. Reliability functions and their characteristics
Title
Reliability
function
failure
rate
hazard rate
mean
Variance
value,
Exponential
e −λt
λ
λe −λt
1
,
λ
1
λ2
µ,
σ2
Normal
¡
¢
Φ µ−t
σ
¡ ¢
Φ σµ
WeibullGnedenko
e −λt
α
−
√
(t−µ)2
2σ 2
−
(t−µ)2
2σ 2
e
e
¢
¡µ¢ √
¡
2πσΦ σ
2πσΦ µ−t
σ
αλt α−1 e −λt
α
αλt α−1
¡
¢
Γ 1 + α1
,
1
λα
¡
¢
¡
¢
Γ 1 + α2 − Γ2 1 + α1
1
λα
Table 5.2. Prolongation
Title
Reliability
function
failure rate
hazard rate
mean
value,
Variance
Gamma
Z
∞
λt
x α−1 −x
(λt)α−1 −λt
λ(λt)α−1 e −λt α ,
R∞
e dxλ
e
Γ(α)
Γ(α)
x α−1 e −λx dx λ
λt
α
λ2
Lognormal
µ
Φ
µ − ln t
σ
¶
(ln t−µ)2
2σ 2
e−
√
2πσt
Reley
e
Uniform
−
t2
2σ 2
t − t 22
e 2σ
σ2
√
e−
(ln t−µ)2
2σ 2
2π σt Φ
1
σ2
³
µ−ln t
σ
e (µ+
´
r
σ2
2
π
σ
2
)
Table 5.3. Damage distributions and their characteristics
Title
Uniform
Shifted
exponential
P.d.f.
Mean
value
1
1{a≤x≤b} ,
b−a
0≤a<b
λe −λ(x−b) 1{x≥b} ,
b > 0, λ > 0
a+b
2
1
b+
λ
Gamma
α α−1
λ x
e −λx 1{x≥0} ,
Γ(α)
α > 0, λ > 0
α
λ
Variance
(b − a)2
12
1
λ2
α
λ2
Table 5.3. Prolongation
Title
P.d.f.
Mean
value
Variance
Lognormal
√
(ln x−µ)2
1
e − 2σ2 1{x≥0} ,
2πσx
e µ+
σ2
2
¡
¢
2
2
e σ − 1 e 2µ+σ
σ>0
Pareto
α ³ c ´α+1
1{x≥c} ,
c x
αc
,
α−1
c > 0, α > 0
α>1
Γ(p + q) p−1
x
(1 − x)q−1 ×
Γ(p) Γ(q)
p
p+q
αc 2
,
(α − 1)(α − 2)
α>2
Beta
pq
(p +
q)2 (p
+ q + 1)
Introduction
4.5. Additions
Exercises.
1. Find c.d.f., p.d.f. the mean value and the variance for an insurance
agreement, presented in the example 1.
2. Let the damage value has an exponential distribution with parameter
λ and a p.d.f.
g (x) = λe −λx 1x≥0 , λ > 0,
Find the m.g.f., the mean value and the variance of that damage.
Introduction
Bibliographical notes
Most of the material of this section in any textbook on Probability
Theory can be found. About ALM-distributions see Chukova, Dimitrov,
(1992), Chukova, Dimitrov, and Khalil, (1993), Dimitrov, Khalil (1992)
Introduction
II.5. Event Tree
In this section the general concepts of hierarchical fault tolerance
reliability systems and the event tree as a tools for their investigation.
Introduction
5.1. Hierarchical Systems
Before turning to Engineering Risk Theory, several words about general
Hierarchical Systems and their analysis tools.
Many of present-day complex technical systems and biological objects are
characterized by the following main properties:
I
I
hierarchical structure;
implemented system of the states control.
Hierarchy of structure means that the system consists of subsystems each
of which is also divided on sub-subsystems etc. up to the lowest
non-divided (elementary) level (see fig.1). We will refer to non-divisible
part of system to as units. Usually hierarchy of structure leads to the
property that the primary failures (faults) arise mainly at the lowest
(elementary) level of the system, and gradually developing leads to the
failure of blocks and subsystems of more higher levels, containing those
elements.
Introduction
®
©
System
­³P ª
³
PP
®
©³
®
©
r r r
S1,1
S1,n1
­©H ª
­©H ª
HH
HH
©©
©©
r r r
r r r
r r r
r
Um
1
r
r
Um
k
Um
K
Subsystems
of level 1
Subsystems
of any level
Units
Fig. 1. A complex multi-level hierarchical system.
Introduction
5.2. Event Tree
Failures of complex systems lead to essential damage, and their
exploitation connects with risks.
Event Tree construction and analysis is an appropriate tools for
hierarchical systems analysis such as degradation and/or fault
development.
Event Tree is a tree-type graph, which root is a resulting event, and
branches connects generating events
Firstly it have been used in Bell Lab by H.A. Watson in the beginning of
60-th last century for system reliability analysis. But this approach could
be used also in many others directions (risk analysis, medicine, etc.)
Introduction
In this course it’ll be used for Engineering Risk Analysis (ERA), and this
means that the failures of appropriate systems will be considered as
events in Event Tree.
This mean that the event are failures of elements, subsystems and the
whole system and the Event Tree will be interpret as a Fault Tree.
For event tree construction due to Henley & Kumamoto (1992) some
special symbols for event and gates are used. They are presented in the
following Tables.
Introduction
Table: Event Symbols
No.
Event symbol
Name
Description
1
Circle
Basic event with sufficient data
2
Diamond
Undeveloped event
3
Rectangle
Event represented by gate
4
Oval
Conditional event used with inhibit gate
5
House
House event. Either occuring or not
6
Triangle
Treansfer symbols
7
Triangle
Treansfer symbols
Introduction
Table: Gate Symbols
No.
Gate
Symbol
Gate
Name
Causal Relation
Function
1
AND
Output event occurs
if all input events occur simultaneously
f =
2
OR
Output event occurs
if any one of the input events occurs
f =1−
3
Inhibit
Input
produces
output when conditional event occurs
f = XU
Q
Xi
Q
(1 − Xi )
Introduction
Table: Gate Symbols. Prolongation
4
Priority
AND
Output event occurs
if all input events occur in the order from
left to right
?
5
Exclusive Output event occurs
OR
if one, but not both,
of the input events
occur
f = X1 (1 − X2 )+
(1 − X1 )X2
m out
of n
f
m
6
n
Output event occurs
if m out of n input
=
αi
P
Qn
αi ≥m
i=1
1−αi
Introduction
This general approach for the Engineering Risk Analysis is used and more
detailed in the next Chapter will be considered.