As discussed above discrete random variables have a finite (or listable) range. What are their probability models? It's easy. In fact, you knew this before taking this course. Right! The probability model of a discrete random variable is its range and associated probabilities.
For example, in the first roll in the game of craps (roll a pair of dice) let X = the sum of the upfaces. Then the range of X is 2, 3, 4, ... ,12. Now ASSUME that the dice are fair. Upon recalling the picture of the sample space, it is easy to determine the probability model of X. For example, the probability that X = 3 means the probability that a (1,2) or a (2,1) comes up which is (1/36) + (1/36) = 2/36. Using the same reasoning for the other range items, we obtain the probability model for X:
Range 2 3 4 5 6 7 8 9 10 11 12 Probabilities 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
Instead of a table, how about a picture of the probability model? Just plot the probabilities (vertical) versus the range (horizontal). Here is a crude plot:
. : . . : : : : : . . : : : : : : : : : . ---+---------+---------+---------+---------+---------+--- 2.0 4.0 6.0 8.0 10.0 12.0Alas, two more items. A little notation here is useful. For a discrete random variable X, let p(x) denote the probability that X assumes the value x. Often p(x) is called the probability mass function. For example, in the dice problem above. We will denote the probability that X is 7 by p(7); hence, p(7) = 6/36.
Note that, in general for any discrete random variable, p(x) is a fraction and the sum of all the p(x) (over the range of X) is 1.
Although the term probability model makes sense here, it is often not used in practice. Usually we call the probability model of X, the distribution of X. It is confusing since we used the term distribution with sampling distributions of Chapter 1. We will sort this out later.