Random Variables and Basic Distributional Quantities · Clearly, the2nd central momentis equal to the variance. Chapters 2 & 3 (Stat 477)Review of Random VariablesBrian Hartman -

Random Variables and Basic Distributional Quantities

Chapters 2 & 3

Stat 477 - Loss Models

Chapters 2 & 3 (Stat 477) Review of Random Variables Brian Hartman - BYU 1 / 29

Introduction

Introduction

Consider an experiment whose outcome is not known in advance.

Sample space Ω is the set of all possible outcomes in the experiment.

A subset E ⊆ Ω of the sample space is called an event.

Complement of an event E, denoted by Ec, is the set of all possibleoutcomes not in E.

Union of two events: E1 ∪ E2 consists of all outcomes that are eitherin E1 or in E2 or in both.

Intersection of two events: E1 ∩E2 (often denoted by E1E2) consistsof all outcomes in both E1 and E2.

Mutually exclusive events: If E1E2 = ∅, i.e. no outcomes in both E1

and E2 so that E1 and E2 cannot both occur, we say they aremutually exclusive events.

These definitions can naturally extend to more than a pair of events.


Introduction axioms

Axioms of probability

Pr(E) will denote a number that is associated with the probability ofthe event E.

Three important axioms of probability:

1 0 ≤ Pr(E) ≤ 1

2 Pr(Ω) = 1 with Ω, the sample space.

3 For any sequence of mutually exclusive events E1, E2, . . .

Pr

( n⋃i=1

Ei

)=

n∑i=1

Pr(Ei), for n = 1, 2, . . . ,∞

Consequence: Pr(Ec) = 1− Pr(E) for any event E.


Introduction conditional probability

Conditional probability

For any two events E and A, we define

Pr(E|A) =Pr(EA)

Pr(A).

Because for any event E, we have E = EA ∪ EAc, then

Pr(E) = Pr(EA) + Pr(EAc)

= Pr(E|A)× Pr(A) + Pr(E|Ac)× Pr(Ac).

Let A1, A2, . . . , An be n mutually exclusive events whose union is thesample space Ω. Then we have the Law of Total Probability:

Pr(E) =

n∑i=1

Pr(EAi) =

n∑i=1

Pr(E|Ai)× Pr(Ai).


Introduction independence

Independence

Two events E1 and E2 are said to be independent if

Pr(E1|E2) = Pr(E1).

As a consequence, we have E1 and E2 are two independent events if

Pr(E1E2) = Pr(E1)× Pr(E2).


Random variables

Random variables

A random variable will be denoted by capital letters: X.

It is a mapping from the sample space to the set of real numbers:X : Ω→ R. The set of all possible values X takes is called thesupport of the random variable (or its distribution).

X is a discrete random variable if it takes either a finite or at mostcountable number of possible values. Otherwise, it is continuous. Acombination of discrete and continuous would be mixed.

Cumulative distribution function (cdf): F (x) = FX(x) = Pr(X ≤ x).

Survival distribution function (sdf): S(x) = SX(x) = Pr(X > x).

Discrete: probability mass function (pmf)p(x) = pX(x) = Pr(X = x).

Continuous: probability density function (pdf)

f(x) = fX(x) =dF (x)

dx.


Random variables

Examples of random variables encountered in actuarialwork

age-at-death from birth

time-until-death from insurance policy issue.

the number of times an insured automobile makes a claim in aone-year period.

the amount of the claim of an insured automobile, given a claim ismade (or an accident occurs).

the value of a specific asset of a company at some future date.

the total amount of claims in an insurance portfolio.


Random variables distribution function

Properties of distribution function

The distribution function must satisfy a number of requirements:

0 ≤ F (x) ≤ 1 for all x.

F (x) is non-decreasing.

F (x) is right-continous, that is, limx→a+ = F (a).

F (−∞) = 0 and F (+∞) = 1.

Some things to note:

For continuous random variables, F (x) is also left-continuous.

For discrete random variables, F (x) forms a step function.

For a mixed random variable, F (x) forms a combination with jumpsand when it does, the value is assigned to the top of the jump.

Because S(x) = 1− F (x), one should be able to deduce properties ofa survival function. See page 14 of Klugman, et al.


Random variables multivariate

Multivariate random variables

Sometimes called random vectors. Notation: (X,Y ).

Joint cdf: F (x, y) = Pr(X ≤ x, Y ≤ y).

Joint sdf: S(x, y) = Pr(X > x, Y > y).

Discrete pmf: p(x, y) = Pr(X = x, Y = y).

Continuous pdf: f(x, y) =∂2F (x, y)

∂x∂y.

For any set of real numbers C and D, we have

P (X ∈ C, Y ∈ D) =

∫∫x∈C,y∈D

f(x, y)dxdy.


Random variables independence

Independent random variables

X and Y are said to be independent if

Pr(X ∈ C, Y ∈ D) = Pr(X ∈ C)× Pr(Y ∈ D).

Also, we have

Pr(X = x, Y = y) = Pr(X = x)× Pr(Y = y)

andf(x, y) = fX(x)× fY (y)

for two independent random variables X and Y .

Note also that if X and Y are independent and so will G(X) andH(Y ).


Random variables hazard function

Hazard function

The hazard function is defined to be the ratio of the density and thesurvival functions:

h(x) = hX(x) =f(x)

S(x)=

f(x)

1− F (x).

It can also be shown that

h(x) = −S′(x)

S(x)= −d logS(x)

dx.

The following relationship immediately follows:

S(x) = exp

(−∫ x

0h(x)dx

).


Random variables expectation

Expectation

Discrete: E[g(X)] =∑

x g(x)p(x)

Continuous: E[g(X)] =∫∞−∞ g(x)f(x)dx

Linearity of expectation: E(aX + b) = aE(X) + b

Special cases:

mean of X: µX = µ = E(X)

variance of X: Var(X) = σ2X = σ2 = E[(X − µX)2].

standard deviation of X: SD(X) = σX = σ = +√

Var(X).

One can show that the variance can also be expressed asVar(X) = E(X2)− [E(X)]2.

The k-th moment of X: µ′k = E(Xk).

The k-th central moment of X: µk = E[(X − µ)k].

Clearly, the 2nd central moment is equal to the variance.


Random variables expectation

Expectation - the multivariate case

Discrete: E[g(X,Y )] =∑

x

∑y g(x, y)p(x, y)

Continuous: E[g(X,Y )] =∫∞−∞

∫∞−∞ g(x, y)f(x, y)dxdy

Covariance of X and Y : Cov(X,Y ) = E[(X − µX)(Y − µY )]. Onecan show that you can also write this asCov(X,Y ) = E(XY )− E(X)E(Y ).

Important property:

Var(X + Y ) = Var(X) + Var(Y ) + 2Cov(X,Y )

If X and Y are independent, then Cov(X,Y ) = 0 so thatVar(X + Y ) = Var(X) + Var(Y ).

Note the converse is NOT always true: if Cov(X,Y ) = 0, then X andY are not necessarily independent.

Correlation between X and Y :

Corr(X,Y ) =Cov(X,Y )√

Var(X)Var(Y )


Random variables other important quantities

Other important summary quantities

Mode of a random variable: the most likely value.

discrete: the value that gives the largest probabilitycontinuous: the value for which the density is largest

coefficient of variation: CV(X) =SD(X)

µX=σ

µdimensionlessuseful for comparing data with different units, or with widely differentmeans.

skewness: γ1 =µ3σ3

for symmetric distributions, γ1 = 0a measure of departure from symmetry

kurtosis: γ2 =µ4σ4

for a Normal distribution, γ2 = 3measures “flatness” or “peakedness” of the distribution


Random variables quantiles

Quantiles

quantile (sometimes called percentile) function: the inverse of thecumulative distribution function.

The 100p-th percentile, for 0 ≤ p ≤ 1, of a distribution is any valueπp satisfying:

F (πp−) ≤ p ≤ F (πp).

the 50-th percentile, π0.5, is the median of the distribution.

The percentile value is not necessarily unique, unless the randomvariable is continuous, in which case, it is the solution to:

F (πp) = p or πp = F−1(p).

In the case it is either discrete or mixed, it is taken to be the smallestsuch possible value:

πp = F−1(p) = infx ∈ R | F (x) ≥ p.


Random variables generating functions

Generating functions

The moment generating function (mgf) of X: MX(t) = E(eXt) forall t this exists.

The probability generating function (pgf) of X: PX(z) = E(zX) forall z this exists.

Note: MX(t) = PX(et) and PX(z) = MX(log(z)).

The mgf generates moments:

MX(t) =

∞∑k=0

1

k!E(Xk)tk

so that E(Xk) =(ddt

)kMX(t)|t=0.

The pgf is generally used for discrete:

PX(z) =

∞∑k=0

zkPr(X = k).


Random variables empirical model

Empirical model

The empirical model to a dataset of sample size n is the discretedistribution that assigns equal probabilities of 1/n to each data point.

If x1, . . . , xn are the data points, then

Pr(X = xi) =1

n, for i = 1, . . . , n.

The empirical cumulative distribution function (ecdf) is defined to be

Fn(x(i)) = Pr(X ≤ x(i)) =i

n, for i = 1, . . . , n,

where x(1), . . . , x(n) are the values of the data in ascending order.

See example on top of page 23.


Excess loss random variable

Excess loss random variableThe excess loss random variable is defined to be Y P = X − d, givenX > d. Its k-th moment can be determined from

ekX(d) = E[(X − d)k | X > d] =

∫∞d (x− d)kf(x)dx

1− F (d), continuous

=

∑xj>d

(xj − d)kp(xj)

1− F (d), discrete

When k = 1, the expected value

eX(d) = E(X − d | X > d)

is called the mean excess loss function. Other names used have been meanresidual life function and complete expectation of life. Using integration byparts, it can be shown:

eX(d) =

∫∞d S(x)dx

S(d)


Left censored and shifted variable

Left censored and shifted random variableThe left censored and shifted random variable is defined to be

Y L = (X − d)+ =

0, X ≤ d,X − d, X > d.

Its k-th moment can be calculated from

E[(X − d)k+] =

∫ ∞d

(x− d)kf(x)dx, continuous

=∑xj>d

(xj − d)kp(xj), discrete

Clearly, we haveE[(X − d)k+] = ekX(d)[1− F (d)]

Note: for dollar events, the distinction between Y P and Y L is one of perpayment versus per loss. See Figure 3.3 on page 26. When k = 1, theexpected value is sometimes called the stop loss premium:

E[(X − d)+] =

∫ ∞d

S(x)dx.


Limited loss variable

Limited loss random variableThe limited loss random variable is defined to be

Y = X ∧ u =

X, X < u,u, X ≥ u.

It is sometimes called the right censored variable. Its k-th moment is

E[(X ∧ u)k] =

∫ u

−∞xkf(x)dx+ uk[1− F (u)], continuous

=∑xj≤u

xkj p(xj) + uk[1− F (u)], discrete

With integration by parts, it can be shown that

E[(X ∧ u)k] = −∫ 0

−∞kxk−1F (x)dx+

∫ ∞0

kxk−1S(x)dx.

Check the case when k = 1. Note the following important relationship:

X = (X − d)+ + (X ∧ d).


Sums of random variables

Sums of random variables

Sometimes referred to as convolutions. Begin with k random variablesX1, . . . , Xk. Its convolution is the sum

Sk = X1 + · · ·+Xk.

One can view the random variables Xi, for i = 1, . . . , k as paymenton policy i, so that the sum Sk refers to the aggregate, or total,payment.

To derive distribution of sums, the assumption of independence of theXi’s is typically made. Then, use the mgf (or pgf) technique:

MSk(t) =

k∏j=1

MXj (t)

or

PSk(z) =

k∏j=1

PXj (z).


Sums of random variables approximating sums

Approximating the distribution of sums of random variables

For approximation purposes, under certain conditions of the first twomoments, we have

limk→∞

Sk − E(Sk)√Var(Sk)

∼ N(0, 1).

This is referred to as the Central Limit Theorem.


Sums of random variables illustrations

Illustrative examples

Show that the sum of independent gamma with the same scale, θ,parameter is another gamma distribution.

Show that the sum of independent Poisson is another Poisson.

Derive the generalized Erlang distribution: If Xi has an exponentialdistribution with mean parameter 1/µi, for i = 1, . . . , n, no two ofwhich have the same mean, then its sum Sn has this distribution.

All these results to be discussed in lecture.


Risk measures

Risk measures

A risk measure is a mapping from the random variable representingloss (risk) to the set of real numbers.

It gives a single value that is intended to provide a magnitude of thelevel of risk exposure.

Notation: ρ(X)

Properties of a coherent risk measure:

subadditive: ρ(X + Y ) ≤ ρ(X) + ρ(Y ).

monotonic: ρ(X) ≤ ρ(Y ) if X ≤ Y for all posssible outcomes.

positive homogeneous: ρ(cX) = cρ(X) for any positive constant c.

translation invariant: ρ(X + c) = ρ(X) + c for any positive constant c.


Risk measures possible uses

Possible uses of risk measures

Premium calculations

Determination of risk/economic/regulatory capital requirements

Reserve calculations

Internal risk management

Financial reporting e.g. meeting regulatory requirements for financialreporting (Basel Accord II)


Risk measures value-at-risk

Value-at-Risk measure

The value-at-risk of X at the 100p% level, denoted by VaRp(X) = πp, isthe 100p-th percentile (or quantile) of the distribution of X and is thesolution to

Pr(X > VaRp(X)) = 1− p.

Some examples:

Normal distribution: VaRp(X) = µ+ σΦ−1(p)

Lognormal distribution: VaRp(X) = exp[µ+ σΦ−1(p)]

Exponential distribution: VaRp(X) = −θ log(1− p)

Pareto distribution: VaRp(X) = θ[(1− p)−1/α − 1]

Some remarks:

Value-at-risk is monotone, positive homogeneous and translationinvariant, but not necessarily subadditive. See Example 3.13.


Risk measures tail VaR

Tail Value-at-Risk measureThe tail value-at-risk of X at the 100p% security level is defined to be

TVaRp(X) = E(X | X > VaRp(X)) =

∫∞πpxf(x)dx

1− F (πp)=

∫∞πpxf(x)dx

1− p.

It is the expected value of the loss, conditional on the loss exceeding thequantile (or VaR). Other formulas for TVaR:

TVaRp(X) =

∫ 1p VaRu(X)du

1−p

TVaRp(X) = VaRp(X) + e(πp)

TVaRp(X) = πp +E(X)−E(X∧πp)

1−pSome remarks:

Other terms used are: conditional tail expectation (CTE), tailconditional expectation (TCE), expected shortfall (ES).

The tail value-at-risk measure is considered coherent.


Risk measures TVaRs for some distributions

Tail value-at-risk for some distributions

Some examples:

Normal: TVaRp(X) = µ+ σφ[Φ−1(p)]

1− p

Lognormal: TVaRp(X) = exp(µ+ σ2/2)Φ(σ − Φ−1(p))

1− pExponential: TVaRp(X) = θ − θ log(1− p) = θ + VaRp(X)

Pareto: TVaRp(X) = VaRp(X) +VaRp(X) + θ

α− 1


Risk measures limitations

Beware of risk measures

Note that a risk measure provides a single value to quantify the levelof risk exposure. There is possibly no single risk measure that canprovide the whole picture of the danger of the exposure.

It is therefore important to be cautious of the uses and limitations ofthe risk measure being used: make sure you understand what the riskmeasure quantifies which aspect of the risk.

Some risk measures are based also on the model being used; bewareof model risk together with the uncertainty of the parameter used inthe model.

Downfall of hedge fund LTCM in 1998: had a sophisticatedVaR-based risk management system in place but errors in parameterestimation, unexpected large market moves, vanishing liquiditycontributed to downfall.


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Random Variables and Basic Distributional Quantities · Clearly, the2nd central momentis equal to the variance. Chapters 2 & 3 (Stat 477)Review of Random VariablesBrian Hartman -