The Sample Description Space of a Random Phenomenon

It has been stated that probability theory is the study of mathematical models of random phenomena; in other words, probability theory is concerned with the statements one can make about a random phenomenon about which one has postulated certain properties. The question immediately arises: how does one formulate postulates concerning a random phenomenon? This is done by introducing the sample description space of the random phenomenon.

The sample description space of a random phenomenon, usually denoted by the letter , is the space of descriptions of all possible outcomes of the phenomenon.

To be more specific, suppose that one is performing an experiment or observing a phenomenon. For example, one may be tossing a coin, or two coins, or 100 coins; or one may be measuring the height of people, or both their height and weight, or their height, weight, waist size, and chest size; or one may be measuring and recording the voltage across a circuit at one point of time, or at two points of time, or for a whole interval of time (by photographing the effect of the voltage upon an oscilloscope). In all these cases one can imagine a space that consists of all possible descriptions of the outcome of the experiment or observation. We call it the sample description space , since the outcome of an experiment or observation is usually called a sample . Thus a sample is something that has been observed; a sample description is the name of something that is observable.

A remark may be in order on the use of the word “space”. The reader should not confuse the notion of space as used in this book with the use of the word space to denote certain parts of the world we live in, such as the region between planets. A notion of great importance in modern mathematics, since it is the starting point of all mathematical theories, is the notion of a set. A set is a collection of objects (either concrete objects, such as books, cities, and people, or abstract objects, such as numbers, letters, and words). A set that is in some sense complete, so that only those objects in the set are to be considered, is called a space. In developing any mathematical theory, one has first to define the class of things with which the theory will deal; such a class of things, which represents the universe of discourse, is called a space. A space has neither dimension nor volume; rather, a space is a complete collection of objects.

Techniques for the construction of the sample description space of a random phenomenon are systematically discussed in Chapter 2 . For the present, to give the reader some idea of what sample description spaces look like, we consider a few simple examples.

Suppose one is drawing a ball from an urn containing six balls, of which four are white and two are red. The possible outcomes of the draw may be denoted by and , and we write or accordingly, as the ball drawn is white or red. In symbols, we write . On the other hand, we may regard the balls as numbered 1 to 6; then we write to indicate that the possible outcome of a draw is a number, 1 to 6.

Next, let us suppose that one draws two balls from an urn containing six balls, numbered 1 to 6. We shall need a notation for recording the outcome of the two draws. Suppose that the first ball drawn bears number 5 and the second ball drawn bears number 3; we write that the outcome of the two draws is . The object is called a 2-tuple. We assume that the balls are drawn one at a time and that the order in which the balls are drawn matters. Then represents the outcome that first ball 3 and then ball 5 were drawn. Further, and represent different possible outcomes. In terms of this notation, the sample description space of the experiment of drawing two balls from an urn containing balls numbered 1 to 6 (assuming that the balls are drawn in order and that the ball drawn on the first draw is not returned to the urn before the second draw is made) has 30 members:

We next consider an example that involves the measurement of numerical quantities. Suppose one is observing the ages (in years) of couples who apply for marriage licenses in a certain city. We adopt the following notation to record the outcome of the observation. Suppose one has observed a man and a woman (applying for a marriage license) whose ages are 24 and 22, respectively; we record this observation by writing the 2-tuple . Similarly, represents the age of a couple in which the man’s age is 18 and the woman’s age is 80. Now let us suppose that the age (in years) at which a man or a woman may get married is any number, 1 to 200. It is clear that the number of possible outcomes of the observation of the ages of a marrying couple is too many to be conveniently listed; indeed, there are possible outcomes! One thus sees that it is often more convenient to describe , rather than to list, the sample descriptions that constitute the sample description space . To describe in the example at hand, we write

We have the following notation for forming sets. We draw two braces to indicate that a set is being defined. Next, we can define the set either by listing its members (for example, and ) or by describing its members, as in (3.2). When the latter method is used, a colon will always appear between the braces. On the left side of the colon, one will describe objects of some general kind; on the right side of the colon, one will specify a property that these objects must have in order to belong to the set being defined.

All of the sample description spaces so far considered have been of finite size. ¹ However, there is no logical necessity for a sample description space to be finite. Indeed, there are many important problems that require sample description spaces of infinite size. We briefly mention two examples. Suppose that we are observing a Geiger counter set up to record cosmic-ray counts. The number of counts recorded may be any integer. Consequently, as the sample description space we would adopt the set of all positive integers. Next, suppose we were measuring the time (in microseconds) between two neighboring peaks on an electrocardiogram or some other wiggly record; then we might take the set {real numbers : of all positive real numbers as our sample description space.

It should be pointed out that the sample description space of a random phenomenon is capable of being defined in more than one way. Observers with different conceptions of what could possibly be observed will arrive at different sample description spaces. For example, suppose one is tossing a single coin. The sample description space might consist of two members, which we denote by (for heads) and (for tails). In symbols, . However, the sample description space might consist of three members, if we desired to include the possibility that the coin might stand on its edge or rim. Then , in which the description represents the possibility of the coin standing on its rim. There is yet a fourth possibility; the coin might be lost by being tossed out of sight or by rolling away when it lands. The sample description space would then be , in which the description denotes the possibility of loss.

Insofar as probability theory is the study of mathematical models of random phenomena, it cannot give rules for the construction of sample description spaces. Rather the sample description space of a random phenomenon is one of the undefined concepts with which the mathematical theory begins. The considerations by which one chooses the correct sample description space to describe a random phenomenon are a part of the art of applying the mathematical theory of probability to the study of the real world.