We have
seen the example
of the uncertain event *a*
= "Spurs win the FA Cup in the year 2011". We can think of
this event as just one state of the *variable** A*
which represents "FA Cup winners in 2011". In this case A
has many states, one for each team entering the FA Cup. We write this as

*A*
= {*a*1,
*a*2,
..., *a*n}

where *a*1
= "Spurs", *a*2
= "Chelsea", *a*3
= "West Ham", etc.

Since in this
case the set *A*
is finite we say that *A*
is a finite discrete variable.

As
another example, suppose we are interested in the number of critical
faults in our control system. The uncertain event is *A*
= "Number of critical faults". Again it is best to think of *A*
as a variable which can take on any of the discrete values
0,1,2,3,... thus

A={0,1,2,3,....;}.

In this case we

Let us define *a*1
as the event "*A*=0",
and *a*2
as the event "*A*=1".

Clearly the
events *a*1
and *a*2
are mutually exclusive and so P(*a*1
or *a*2)=P(*a*1)+P(*a*2).
However, we cannot say that

P(*a*1
or *a*2)
= 1

because *a*1
and *a*2
are not *exhaustive*.
That is, they do not form a complete partition of *A*.
However, if we define *a*3
as the event "*A*>1"
then *a*1,
*a*2,
and *a*3
are complete and mutually exhaustive and in this case

P(*a*1)+P(*a*2)+P(*a*3)
= 1

In general if A
is a variable with states *a*1,
*a*2,
..., *a*n:

The *probability
distribution *of
*A*,
written P(*A*),
is simply the set of values {P(*a*1),
P(*a*2),
..., P(*a*n)}