Random Variables from the Point of View of a General Theory of Variables

Abstract

In his great book Sequential Analysis Wald defines (see p. 5 in [1]) a random variable as a variable x such that “for any given number c a definite probability can be ascribed to the event that x will take a value less than c ” As a first example of a random variable, Wald mentions the outcome x of the experiment of weighing an object selected at random from a lot of n known objects. He calls x a random variable “since a probability can be ascribed to the event that x will take a value less than c for any given c.” If n _c is the number of objects in the lot whose weight is less than c that probability is n _c/n. On page 11, Wald says that “statistical problems arise when the distribution function of a random variable is not known and we want to draw some inference concerning the unknown distribution function on the basis of a limited number of observations.” He then mentions, as an example, the random variable x assuming the value 0 if a unit selected from a completely unknown lot of products is nondefective, and the value 1 if the unit is defective.