Random Variables and Probability Distributions

Abstract

When the average citizen sees a game of dice, he knows intuitively that the outcome of any 1 roll of the dice is unpredictable—that he is faced with a “chance” or “random” phenomenon. He sees quickly that there are 36 possible outcomes (for each of the 6 sides of die #1 he can get any of the 6 sides of die #2). He nevertheless realizes the possibilities of betting on the outcome when he sees that there is only 1 outcome, (1,1), which gives a “2,” while 6 outcomes, (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), give a “7.” Thus the “probability” of a 2 is 1/36 and that of a 7 is 6/36. The set of outcomes or “scores” with their associated probabilities constitutes a probability distribution and is his best aid to intelligent betting. In that case the dice were treated or “shaken” alike. Individual outcomes differ, but in the “long-run” one can predict how often each outcome will occur. What the layman may not realize is that even in a very precise chemical experiment the outcome is random. The possible outcomes may lie within a narrow range, but when seemingly identical units are treated as much alike as possible, they still respond differently; there is still some “variability” in the outcome, and the chemist, with the aid of statistics, summarizes the data by telling his readers what they can bet on. We shall now repeat those ideas with more detail.