Random variable

           A random variable associates the points in the sample space with real numbers.

           Consider the probability space and function mapping the sample space into the real line. Let us define the probability of a subset by

                                           

           Such a definition will be valid if is a valid event. If is a discrete sample space, is always a valid event, but the same may not be true if is infinite. The concept of sigma algebra is again necessary to overcome this difficulty. We also need the Borel sigma algebra -the sigma algebra defined on the real line.

           The function is called a random variable if the inverse image of all Borel sets under is an event. Thus, if is a random variable, then