Random Variable It is often desirable to assign numbers to the nonnumerical outcomes of a sample space. This assignment leads to the concept of a random variable. A random variable X is a set function which assigns to each outcome EÎS a real number X(E)=x. The domain of this set function is the sample space S and its range is the set of real numbers. In general, a random variable has some specified physical, geometrical, or other significance. It is important to observe the difference between the random variable itself and the value it assumes for a typical outcome. It has become common practice to use capital letters for a random variable and small letters for the numerical value that the random variable assumes after an outcome has occurred.
Informally speaking, we may say that a random variable is the name we give an outcome before it has happened. It may be that the outcomes of the sample space are themselves real numbers, such as when throwing a die. In such a case, the random variable is the identity function X(E)=E. Note that, strictly speaking, when we are throwing a die the outcomes are not numerical, but are
the “dot patterns” on the top face of the die. It is, of course, quite natural, but really not necessary, to associate the face values with the corresponding real number.
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