# Variables and probability distributions

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 = {a1, a2, ..., an}

where a1 = "Spurs", a2 = "Chelsea", a3 = "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 a1 as the event "A=0", and a2 as the event "A=1".

Clearly the events a1 and a2 are mutually exclusive and so P(a1 or a2)=P(a1)+P(a2). However, we cannot say that

P(a1 or a2) = 1

because a1 and a2 are not exhaustive. That is, they do not form a complete partition of A. However, if we define a3 as the event "A>1" then a1, a2, and a3 are complete and mutually exhaustive and in this case

P(a1)+P(a2)+P(a3) = 1

In general if A is a variable with states a1, a2, ..., an:

The probability distribution of A, written P(A), is simply the set of values {P(a1), P(a2), ..., P(an)}