## Random Variables

A variable whose value is determined by the outcome of a chance experiment is called a random variable (rv). Random variables are usually denoted by the capital letters X, Y, Z, and so on, and the values taken by them are denoted by small letters x, y, z, and so on.

A random variable may be either discrete or continuous. A discrete rv takes on only a finite (or countably infinite) number of values.2 For example, in throwing two dice, each numbered 1 to 6, if we define the random variable X as the sum of the numbers showing on the dice, then X will take one of these values: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12. Hence it is a discrete random variable. A continuous rv, on the other hand, is one that can take on any value in some interval of values. Thus, the height of an individual is a

'A function whose domain and range are subsets of real numbers is commonly referred to as a real-valued function. For details, see Alpha C. Chiang, Fundamental Methods of Mathematical Economics, 3d ed., McGraw-Hill, 1984, Chap. 2.

2For a simple discussion of the notion of countably infinite sets, see R. G. D. Allen, Basic Mathematics, Macmillan, London, 1964, p. 104.

872 APPENDIX A: A REVIEW OF SOME STATISTICAL CONCEPTS

continuous variable—in the range, say, 60 to 65 inches it can take any value, depending on the precision of measurement.

A.4 PROBABILITY DENSITY FUNCTION (PDF)

Probability Density Function of a Discrete Random Variable

Let X be a discrete rv taking distinct values x1, X2, •••, , •••• Then the function f (x) = P(X = xi) for i = 1,2,. . . , n,... = 0 for x = Xi is called the discrete probability density function (PDF) of X, where P(X = xi) means the probability that the discrete rv X takes the value of xi. 