Probability

Values of a Variable

By definition, the values that a variable can take must be both exclusive and exhaustive. Exclusive means that two values cannot be true simultaneously. Exhaustive means that the values must cover all possible cases. For example, the values male and female are mutually exclusive because a person cannot be a man and a woman; they are exhaustive because there exists no other possibility.

Types of Variables

A variable is said to be discrete if its domain has a finite number of values. In the above example, Sex takes on only two values. A variable is said to be continuous if its domain is a numerical interval, such as [0,1] or [−∞,∞]. For instance, age, weight, height, temperature, red cell count, end-diastolic area of a valve, and so on are all continuous variables.

A continuous variable can be discretized by partitioning its domain into a finite number of subintervals. For instance, when modeling a medical problem, we could define three intervals for the variable Age: young = from 0 to 25, adult = from 26 to 70, and elderly = over 70. In another situation, it might be more appropriate to define the intervals differently, for example, young = from 0 to 15, adult = from 16 to 65, and elderly = over 65, or even to define more intervals: from 0 to 5, from 6 to 10, from 11 to 15, from 16 to 20, and so on.

Probability of a Discrete Variable

As a first approach, we can define the probability of a discrete variable X as a function that assigns to each value x a number between 0 and 1 (both inclusive) such that the sum of them must be 1:

For example, for the variable Age mentioned above, we could have the following assignment of probability: P(young) = .35, P(adult) = .46, and P(elderly) = .19. Each of these probabilities is between 0 and 1, and the sum of all is 1: P(young) + P(adult) + P(elderly) = 1.

Probability of a Continuous Variable

The definition of the probability of a continuous variable is much more complex than in the discrete case. Let us assume that X is a continuous variable taking on real values. The basis for the definition of a continuous probability distribution is a function F(x), called a cumulative distribution function, which, by definition, must satisfy the following properties.

Roughly speaking, the first expression means that F(x) increases—or at least does not decrease— when x increases, and the latter two mean, respectively, that the smaller the value of x, the closer is F(x) to 0 and the greater the value of x, the closer is F(x) to 1 (see Figure 1).

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