We now define the concept of a random variable, a key definition in probability and statistics and for statistical simulation in general. We define a random variable as a real number that is assigned to an event . It is random because the event is random, and it is variable because the assignment of the value may vary over the real axis. We will use ``r.v.'' as an abbreviation for ``random variable''.
Assign the number 10n to each face n of a die. When face n appears, the r.v. is 10n.
Random variables are useful because they allow the quantification of random processes, and they facilitate numerical manipulations, such as the definition of mean and standard deviation, to be introduced below. For example, if one were drawing balls of different colors from a bowl, it would be difficult to envision an ``average'' color, although if numbers were assigned to the different colored balls, then an average could be computed. On the other hand, in many cases of real interest, there is no reasonable way to assign a real number to the outcome of the random process, such as the outcome of the interaction between a 1 eV neutron and a uranium-235 nucleus, which might lead to fission, capture, or scatter. In this case, defining an ``average'' interaction makes no sense, and assigning a real number to the random process does not assist us in that regard. Nevertheless, in the following discussion, we have tacitly assumed a real number has been assigned to the event that we know occurs with probability . Thus, one can in essence say that the r.v. occurs with probability .