Discrete Random Variables

So far, our examples of random variables have been discrete random variables. H or T could only take on discrete values on any specific toss: 1 or 0. On any given toss, we have no way of knowing what value H or T will take, but we can estimate or calculate what the probable outcomes are, and we can say for certain, because the random variables are discrete, that they will not take on in-between values. For sure, H cannot take on a value of 0.75 on any specific toss; only values of 1 (true) or 0 (false) are allowed. Sometimes knowing what values cannot happen is as important as knowing what values will happen.

Random variables are quite useful in projects when counting things that have an atomic size. People, for instance, are discrete. There is no such thing as one-half a person. Many physical and tangible objects in projects fit this description. Sometimes actions by others are discrete random variables in projects. We may not know at the outset of a project if a regulation will be passed or a contract option exercised, but we can calculate the probability that an action will occur, yes or no.

Many times there is no limit to the number of values that random variables can take on in the allowed range. There is no limit to how close together one value can be to the next; values can be as arbitrarily close together as required. The only requirement is that for any and all values of the discrete random variable, the sum of all their probabilities of occurrences equals 1:

∑ all f_i(X | a_i) = 1, for i = 1 to "n"

where f_i(X) is one of "n" probability functional values for the random variable, there being one functional value for each of the "n" values that X can take on in the probability space, and "a_i" is the ith probable value of X.