ABSTRACT
In real-world problems we are often faced with one or more quantities that do not have fixed values. The values of such quantities depend on random actions, and they usually change from one experiment to another. For example, the number of babies born in a certain hospital each day is not a fixed quantity. It is a complicated function of many random factors that vary from one day to another. So are the following quantities: the arrival time of a bus at a station, the sum of the outcomes of two dice when thrown, the amount of rainfall in Seattle during a given year, the number of earthquakes that occur in California per month, and the weight of grains of wheat grown on a certain plot of land (it varies from one grain to another). In probability, quantities introduced in these diverse examples are called random variables. The numerical values of random variables are unknown. They depend on random elements occurring at the time of the experiment and over which we have no control. For example, if in rolling two fair dice, X is the sum, then X can only assume the values 2, 3, 4, . . . , 12 with the following probabilities:
P (X = 2) = P ({
(1, 1) })
= 1/36,
P (X = 3) = P ({
(1, 2), (2, 1) })
= 2/36,
P (X = 4) = P ({
(1, 3), (2, 2), (3, 1) })
= 3/36,
and, similarly,
Sum, i 5 6 7 8 9 10 11 12
P (X = i) 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
Clearly, {2, 3, 4, . . . , 12} is the set of possible values of X . Since X ∈ {2, 3, 4, . . . , 12}, we should have
∑12 i=2 P (X = i) = 1, which is readily verified. The numerical value of
a random variable depends on the outcome of the experiment. In this example, for instance,
if the outcome is (2, 3), then X is 5, and if it is (5, 6), then X is 11. X is not defined for points that do not belong to S, the sample space of the experiment. Thus X is a real-valued function on S. However, not all real-valued functions on S are considered to be random variables. For theoretical reasons, it is necessary that the inverse image of an interval in R be an event of S, which motivates the following definition.
