L4 C1 S1 Definition and Intuition

Remember that A Model is never Real -- it is an ideal which has existence in our minds. These ideals need not have any correspondent in reality. The continuous random variable is such a construct, where the basic concept is unlikely to find any counterpart in reality. Nonetheless it is a useful idealization. We consider a CONTINUUM of possible outcomes -- all real numbers from [-a,+b]. ALL of these are possible outcomes. Now the problem is that if all of these outcomes are assigned probabilities, then no matter how we do it, the sum of all the probabilities will become infinite. So the solution is to use INFINITESMAL probability. We will use dx to denote an infinitesmal real number. This dx is smaller than 1/n for all integers n, but it is bigger than 0. There are many details about how this can be constructed, and how it has to be the RIGHT SIZE of infinitesmal to work in defining probability -- even among infinitesmals, there are different sizes, some of which will be too small while others will be too large. HOWEVER< we ignore all this, and proceed in an intuitive and informal way.