P&S L2 C1 S1

DEFINITION: Consider a HYPOTHETICAL population, which is subdivided into two types of members -- suppose that M members belong to category A, and N members belong to category B. Let S be a simple random draw from this population. Define random variable X(S) to equal 0 if the simple random draw S belongs to category A, and define X(S) to be 1 if the simple random draw belongs to category B. Then X(S) is a Bernoulli Random Variable, which takes value 0 with probability M/(M+N) and 1 with probability N/(M+N).

IMPORTANT NOTE: This definition creates a random variable in an imaginary world, where a perfect simple random draw is made from an imaginary and idealized population consisting of an exactly known collection of members. This serves as a useful model for many real world situations, where we identify the hypothetical population with the real population. Thinking and calculations are simplified by pretending that the MODEL is the REALITY, but for some purposes it is essential to keep the two separate.

Reality need not match the model. It is essential to keep the two separate, but it is easy to confuse the two, and think of the model as the reality.