The Same Example Treated From the Point of View of Random Variables

We now treat the example considered in the foregoing section in terms of random variables. We shall see that the notion of a random variable does not replace the idea of a numerical valued random phenomenon but rather extends it.

We let and denote, respectively, the departure time of the train and the arrival time of the commuter at the station. In order, with complete rigor, to regard and as random variables, we must state the probability space on which they are defined as functions. Let us first consider . We may define as the identity function (so that , for all in ) on a real line , on which a probability distribution (that is, a distribution of probability mass) has been placed in accordance with the probability density function given by (3.1) . Or we may define as a function on a space of 2-tuples of real numbers, on which a probability distribution has been placed in accordance with the probability density function given by (3.12) ; in this case we define . Similarly, we may regard as either the identity function on a real line , on which a probability distribution has been placed in accordance with the probability density function given by (3.2) , or as the function with values , defined on the probability space . In order to consider and in the same context, they must be defined on the same probability space. Consequently, we regard and as being defined on .

It should be noted that no matter how and are defined as functions the individual probability laws of and are specified by the probability density functions and , with values at any real number ,

Consequently, the random variables and are identically distributed.

We now turn our attention to the problem of computing the probability that the man will catch the train. In the previous section we reduced this problem to one involving the computation of the probability of a certain event (set) on a probability space. In this section we reduce the problem to one involving the computation of the distribution function of a random variable; by so doing, we not only solve the problem given but also a number of related problems.

Let denote the difference between the train’s departure time and the man’s arrival time . It is clear that the man catches the train if and only if . Therefore, the probability that the man will catch the train is equal to . In order for to be a meaningful expression, it is necessary that be a random variable, which is to say that is a function on some probability space. This will be the case if and only if the random variables and are defined as functions on the same probability space. Consequently, we must regard and as functions on the probability space , defined in the second paragraph of this section. Then is a function on the probability space , and is meaningful. Indeed, we may compute the distribution function of , defined for any real number by

Then .

To compute the distribution function of , there are two methods available. In one method we use the fact that we know the probability space on which is defined as a function and use (4.2) . A second method is to use only the fact that is defined as a function of the random variables and . The second method requires the introduction of the notion of the joint probability law of the random variables and and is discussed in the next section. We conclude this section by obtaining by means of the first method.

As a function on the probability space is given, at each 2-tuple , by . Consequently, by (4.2) , for any real number ,

From (4.3) we obtain an expression for the probability density function of the random variable . In the second integration in (4.3), make the change of variable . Then

By interchanging the order of integration, we have

By differentiating the expression in (4.4) with respect to , we obtain the integrand of the integration with respect to , with replaced by ; thus

Equation (4.5) constitutes a general expression for the probability density function of the random variable defined on a space of 2-tuples by , where a probability function has been specified on by the probability density function .

To illustrate the use of (4.5), let us consider again the probability density functions introduced in connection with the problem of the commuter catching the train. The probability density function is given by (3.12) in terms of the functions and , given by (3.1) and (3.2), respectively.

In the case of independent phenomena, (4.5) becomes

If further, as is the case here, the two random phenomena [with respective probability density functions and ] are identically distributed, so that, for all real numbers , for some function , then the probability density function is an even function; that is, for all . It then suffices to evaluate for . One obtains, by using (3.1), (3.2), and (4.6), Therefore, Consequently

Exercises