What is a random variable ? DA Freedman

Suppose you reach into your pocket and pull out a coin; think of one side as “heads” and the other as “tails.” You toss the coin three times. We could now define some random variables, at least informally, e.g., • the number of heads on the first toss (which is 0 or 1), • the total number of heads (which is 0, 1, 2, or 3). This experiment is a little hypothetical—hence the subjunctive voice. If we actually did it and got “head tail head,” the observed value of our first random variable—the number of heads on the first toss—would be 1; the observed value of the other random variable—the total number of heads—would be 2. Even after the experiment is done and the data are collected, we can still think about the things that could have happened but didn’t. Random variables and probabilities are mainly not about actual outcomes. Rather, these concepts apply to an experiment and the various ways it could have turned out. In fact, most statistical calculations (e.g., confidence levels and significance levels) apply to the experiment and all of its possible outcomes, including the ones that did not materialize. Reaching into pockets and pulling out coins are not mathematical ideas. The connection between real coins and mathematical models is a complicated topic. To continue the discussion, we need to start from another angle. A.N. Kolmogorov laid the foundations for modern probability theory with the “sample space representation” of probabilities. For the coin, this amounts to a list of the 23 = 8 possible outcomes: