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326 13 Inductive vs. Deductive Reasoning

1. The probability on intervals or regions. For example, a temperatureT
   defines events such as(t 1 ≤T≤t 2 ), the set of temperatures betweent 1
   andt 2.


2. The ratio of the probability on a small interval or region divided by the
   size of the interval or region. This is called theprobability density.

As with discrete random variables, the total probability must be 1. In terms

of the probability density this means that the integral over all possible values

is equal to 1.

Both discrete and continuous random variables have total probability 1.

However, it is sometimes convenient to relax this requirement. An assign-

ment of nonnegative weights to the possible values of a variable is called

adistribution. In mathematics, a distribution is called ameasure. If a distri-

bution has a total weight that is some positive number other than 1, then

one can convert the distribution to a probability distribution by dividing all

weights by the total weight. This is callednormalization. However, some

distributions cannot be normalized because their total weight is either 0 or

infinite.

The simplest example of a distribution is theuniform distribution. A ran-

dom variable has the uniform distribution when every value has the same

weight as every other value. A continuous random variable is uniform when

its density is a positive constant. Uniform distributions on infinite sets or

regions cannot be normalized.

When there are several random variables, their probabilistic structure is

completely defined by the intersections of their events. Thus the events de-

fined by the random variablesLandTinclude such events as

(L=greenandT≤5).

The probabilities of these events define thejoint probability distribution(JPD)

of the random variablesLandT.Astochastic modelis another name for a

collection of random variables. The probabilistic structure of the stochastic

model is the JPD of the collection of random variables. One could give a

strong argument that the stochastic model is the fundamental construct, and

that the probability space is secondary. However, it is convenient to treat the

probability space as fundamental and the random variables as derived from

it (as measurable functions on the probability space).

Given two eventsAandB,aconditional probabilityofAgivenBis any

numbercbetween 0 and 1 (inclusive) such thatPr(AandB)=cP r(B).