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This function is graphed in Figure 4.2.

The probability function of a discrete random variable must satisfy the two conditions given in the box.

Properties of Probability Functions of Discrete Random Variables

Let X be a discrete random variable with probability function Px(x). Then

(ii) The individual probabilities sum to 1; that is

where the notation indicates summation over all possible values x.

Property (i) merely states that probabilities cannot be negative. Property (ii) follows from the fact that the events "X = x" for all possible values x, are mutually exclusive and collectively exhaustive. The probabilities for these events must therefore sum to 1. That this is in fact so for Examples 4.1 and 4.2 can be verified directly. It is simply a way of saying that when a random experiment is to be carried out, something must happen.

FIGURE 4.1 Probability function for Example 4.

Another representation of discrete probability distributions is also useful.

Definition

The cumulative probability function, F*(x0), of a random variable X expresses the probability that X does not exceed the value x,\, as a function of That is fx(xo) = P(X < where the function is evaluated at all values x0.

For discrete random variables, the cumulative probability function is sometimes called the cumulative mass function. It can be seen from the definition that, as x0 increases, the cumulative probability function will change values only at those points x{) that can be taken by the random variable with positive probability. Its evaluation at these points can be carried out in terms of the probability function.

FIGURE 4.2 Probability function for Example 4.2 PxM