This unit introduces over 40 commonly used probability distributions. 

Discrete and Continuous distributions 

A random variable is discrete if it can assume at most a finite or countably infinite number of possible values. A random variable is continuous if it can assume any value in some interval or intervals of real numbers and the probability that it assumes any specific value is zero. 

PDF, CDF and inverse CDF 

When dealing with the random variable, it is not enough just to determine what values are possible. There is also a need to determine what is probable. We must be able to predict in some sense the values that the variable is likely to assume at any time. Since the behavior of a random variable is governed by chance, these predictions must be made in the face of a great deal of uncertainty. The best that can be done is to describe the behavior of the random variable in terms of probabilities. Two functions are used to accomplish this, probability density functions and cumulative distribution functions. 

Probability density function (PDF) 

Discrete distributions: There are several facts to note concerning the PDF in the discrete case. First, the PDF is defined on the entire real line, and for any given real number x, f(x) is the probability that the random variable X assumes the value x. The necessary and sufficient conditions for a function f to be a PDF are: 

Even though a discrete density is defined on the entire real line, it is only necessary to specify the density for those values x for which f(x) is not zero. 

Continuous distributions: The PDF is defined on an entire real line and is nonnegative. In the continuous case the probability that the random variable X assumes the value x is 0. Therefore, we are interested in finding the probability that random variable X assumes values in some interval [a,b]. The necessary conditions for a function f to be a PDF are: 

In Probabilities.pas all probability density functions have a PDF at the end of their name. 

Cumulative distribution function (CDF) 

Discrete distributions: The word "cumulative" suggests the role of the CDF function. It sums or accumulates the probabilities found by means of PDF. The cumulative distribution function F(x) is defined as follows: 

We must sum the PDF over all values of X that occur with nonzero probability that are less than or equal to x. 

Continuous distributions: In continuous case is the cumulative distribution function - CDF defined exactly as in the discrete case although found by using integration rather than summation. The cumulative distribution function F(X) is defined as follows: 

In this case, to find a specific F(x), we must integrate the density over all real numbers that are less than or equal to x. In Probabilities.pas all cumulative distribution functions have a CDF at the end of their name. 

Getting the "original values" from given probability (inverse CDF) 

Generally, the "inverse" cumulative distributions are used to "get" the original values that were used for calculating the CDF. If the distribution is continuous, the result of the inverse CDF is the original value. If the distribution is discrete, the result is generally not the exact original value. 

Discrete distributions: As stated above, the relationship between a discrete CDF and its inverse function is more complicated. It is likely that there is no such value x that the CDF of x gives the desired probability. In these cases the inverse function returns the first value x such that the CDF of that value equals or exceeds the desired probability. 

Continuous distributions: Calculating the CDF of values gives the probabilities in the interval [0,1]. If we want the original values that were used for the CDF (x), we must apply the inverse cumulative distribution. The result is the original value used to calculate CDF. In Probabilities.pas all inverse cumulative distribution functions have a CDFInv at the end of their name.