Risk Analysis in Finance and Insurance, 2nd Edition

6 Risk Analysis in Finance and Insurance

Thetriple(Ω, F,P) is called a probability space. For the rest of this section,

we assume that the set Ω is countable. In this case, (Ω, F,P) is referred to as

a discrete probability space.

Every event A ∈Fcan be associated with its indicator:

I

A

(ω)=



1 , if ω ∈ A

0 , if ω ∈ Ω \A

.

Any function X :Ω→ R is called a random variable. An indicator is the

simplest example of a random variable. Any random variable X on a discrete

probability space is discrete since the range of function X(·) is countable:

(x

k

)

k=1,2,...

. In this case, we have the following representation:

X(ω)=

∞



k=1

x

k

I

A

k

(ω) ,

where A

k

∈Fand ∪

k

A

k

= Ω. A discrete random variable X is called simple

if the corresponding sum is finite. The function

F

X

(x):=P ({ω