PROBABILITY THEORY:RANDOM VARIABLES

RANDOM VARIABLES

It may occur frequently that in performing an experiment one is not interested directly in the entire sample space or in events defined over the sample space. For example, suppose that the experiment which measures the times of the first arrival on 2 days was performed to determine at what time to open the store. Prior to performing the experiment, the store owner decides that if the average of the arrival times is greater than an hour, thereafter he will not open his store until 10 A.M. (9 A.M. being the previous opening time). The aver- age of x1 and x2 (the two arrival times) is not a point in the sample space, and hence he cannot make his decision by looking directly at the outcome of his experiment. Instead, he makes his decision according to the results of a rule that assigns the average of x1 and x2 to each point (x1,x2) in n. This resultant set is then partitioned into two parts: those points below 1 and those above 1. If the observed result of this rule (average of the two arrival times) lies in the partition with points greater than 1, the store will be opened at 10 A.M.; otherwise, the store will continue to open at 9 A.M. The rule that assigns the average of x1 and x2 to each point in the sample space is called a random variable. Thus, a random variable is a numerically valued function defined over the sample space. Note that a function is, in a mathematical sense, just a rule that assigns a number to each value in the do- main of definition, in this context the sample space.

Random variables play an extremely important role in probability theory. Experiments are usually very complex and contain information that may or may not be superfluous. For example, in measuring the arrival time of the first customer, the color of his shoes may be pertinent. Although this is unlikely, the prevailing weather may certainly be rele- vant. Hence, the choice of the random variable enables the experimenter to describe the factors of importance to him and permits him to discard the superfluous characteristics that may be extremely difficult to characterize.

There is a multitude of random variables associated with each experiment. In the ex- periment concerning the arrival of the first customer on each of 2 days, it has been pointed out already that the average of the arrival times X is a random variable. Notationally, random variables will be characterized by capital letters, and the values the random variable takes on will be denoted by lowercase letters. Actually, to be precise, X should be writ- ten as X(w), where w is any point shown in the square in Fig. 24.1 because X is a function. Thus, X (1,2) = (1 + 2)/2 = 1.5, X (1.6,1.8) = (1.6 + 1.8)/2 = 1.7, X (1.5,1.5) = (1.5 + 1.5)/2 = 1.5, X(8,8) = (8 + 8)/2 = 8. The values that the random variable X takes on are the set of values x such that 0 ::: x ::: 8. Another random variable, X1, can be de- scribed as follows: For each w in n, the random variable (numerically valued function) disregards the x2 coordinate and transforms the x1 coordinate into itself. This random vari- able, then, represents the arrival time of the first customer on the first day. Hence, X1(1,2) = 1, X1(1.6,1.8) = 1.6, X1(1.5,1.5) = 1.5, X1(8,8) = 8. The values the random variable X1 takes on are the set of values x1 such that 0 ::: x1 ::: 8. In a similar manner, the random variable X2 can be described as representing the arrival time of the first customer on the second day. A third random variable, S2, can be described as follows: For each w in n, the random variable computes the sum of squares of the deviations of the coordinates about their average; that is, S2(w) = S2(x1, x2) = (x1 - x)2 + (x2 - x)2. Hence, S2(1,2) = (1 - 1.5)2 + (2 - 1.5)2 = 0.5, S2(1.6,1.8) = (1.6 - 1.7)2 + (1.8 - 1.7)2 = 0.02, S2(1.5,1.5) = (1.5 - 1.5)2 + (1.5 - 1.5)2 = 0, S2(8,8) = (8 - 8)2 + (8 - 8)2 = 0. It is evident that the values the random variable S2 takes on are the set of values s2 such that 0 ::: s2 ::: 32.

All the random variables just described are called continuous random variables be- cause they take on a continuum of values. Discrete random variables are those that take on a finite or countably infinite set of values.1 An example of a discrete random variable can be obtained by referring to the experiment dealing with the measurement of demand. Let the discrete random variable X be defined as the demand during the month. (The experiment consists of measuring the demand for 1 month). Thus, X(0) = 0, X(1) = 1, X(2) = 2, . . . , so that the random variable takes on the set of values consisting of the integers. Note that n and the set of values the random variable takes on are identical, so that this random variable is just the identity function.

From the above paragraphs it is evident that any function of a random variable is it- self a random variable because a function of a function is also a function. Thus, in the pre- vious examples X = (X1 + X2)/2 and S2 = (X1 - X)2 + (X2 - X)2 can also be recognized as random variables by noting that they are functions of the random variables X1 and X2.

This text is concerned with random variables that are real-valued functions defined over the real line or a subset of the real line.