3 - DISCRETE DISTRIBUTIONS It is always helpful when solving a problem to be able to relate it to a problem whose solution is already known and understood. In probability theory, many problems turn out to be special cases of standard examples. The most common standard examples are the well-known distributions. Discrete Distributions In simple terms, a distribution is an indexed set of probabilities whose sum is 1. For the moment, discussion will be restricted to cases where there is a single discrete random variable X whose value r runs from zero upwards and serves as the index. It is possible to think of r running from 0 to oo where in most cases the indexed probability is zero. A distribution may be expressed by a table or a function or graphically. Consider the distribution associated with a fair die. A tabular representation of the distribution is: r -> X 0 P(X=r) 0 6 1 2 1 6 1 6 3 4 5 6 1 6 1 1 6 6 1 6 A functional representation of the distribution is: 1 6' P(X = r) = - ~ 0, otherwise ifrENA16r66 A graphical representation of the distribution is: 1 1 6 P(X=r) 0 6 0 1 2 3 4 5 6 r -> Of the three representations, only the function makes it pedantically clear that unless 1 6 r 6 6 the probability is zero. - 3.1 -
The Uniform Distribution When all the non-zero probabilities are the same and are indexed by a contiguous sequence of values of r, the distribution is said to be a Uniform distribution. The behaviour of a fair die is an example of a Uniform distribution. All six non-zero probabilities are the same and the index r for these probabilities has the contiguous values 1, 2, 3, 4, 5 and 6. One can imagine a fair die with a different number of faces. Consider a fair tetrahedral die whose faces happen to be numbered 5, 6, 7 and 8. The four probabilities are all Z. There is actually a family of distributions and the description: Uniform(m, n) is used to refer to the general case; m and n are the start and stop values of r and are called the parameters of the distribution. A random variable whose value represents the outcome of throwing an ordinary fair die is said to be 'distributed Uniform(1,6)'. If the value represents the outcome of throwing the curious tetrahedral die the random variable is distributed Uniform(5,8). In the general case there are n - m + 1 values for the index and, given the equiprobable nature of the Uniform distribution, the probabilities are all 1/(n-m +1). The functional representation of the general case is: P(X=r) ={n-m+1' 1 ifrENAm 6 r 6 n 0, otherwise It is good practice always to check that the probabilities in a distribution sum to 1. With the general Uniform distribution the check is straightforward: n X n-m+ 1 r=m 1 n-m+1 n-m+1 = 1