ABSTRACT
We saw in Chapter 2 that probability is a function that maps an event in sample space to a number between 0 and 1. The random variable is the tool we need to describe the set on which the probability function operates. The values that a random variable takes are associated with the outcomes of an experiment. For example when a fair die is rolled, then any of the numbers 1, 2,…6 may appear and the random variable X may be defined as the number of 5’s that may show up in 10 rolls of the die. In this case the sample space S will consist of all possible outcomes in the rolling of a die 10 times. There will in fact be a total of 610 elements in S since each roll of a die may result in 1 or 2 or.…6. The outcome of interest in this experiment may be whether there were zero 5’s or one 5 or two 5’s…or ten 5’s. Hence we may define X as a random variable which takes values 0,1,…,10 with probabilities p0, p 1.…, p 10 where p i is the probability that in 10 rolls of a die there are exactly i 5’s. A function X defined on a sample space S together with a probability function P on S is called a random variable. Since the random variables are based on the elements of the sample space S and the probability functions are based on the subsets of the sample space the probability function may be perceived to operate on X. Therefore an intelligent selection of a random variable will lead to the identification of probability for events of interest in S if the values that X takes is a subset s of S such that s ∈ S. In this case the function X transforms points of S into points on the x axis and we say that the probability function operates on X.
