Usage Notes

The distribution function for the (real, single-valued) random variable X is the function F defined for all

real x by

\[F(x) = \Pr(X ≤ x)\]

where Pr(⋅) denotes the probability of an event. The distribution function is often called the cumulative distribution function (CDF).

In general, the CDF does not have an inverse because it is not one-to-one. Nevertheless, a quantile function, also called an inverse CDF, is well-defined as

\[F^{-1}(p) = \min \{x∈ ℜ : F(x)≥p\}, p ∈ (0,1)\]

Here p represents a probability on the open interval, (0, 1).

Definitions and discussions of the terms basic to this chapter can be found in Johnson and Kotz (1969, 1970a, 1970b). These are also good references for the specific distributions.

Discrete Distributions

For discrete distributions, the function giving the probability that a random variable takes on specific values is called the probability mass function, or just probability function, defined by

\[f(x) = \Pr(X = x)\]

The CDF for a discrete random variable is

\[F(x) = \Pr[X \leq x] = \sum_A f(k)\]

where A is the set such that \(k \leq x\).

Continuous Distributions

For continuous distributions, a probability function, as defined above, would not be useful because the probability of any given point is 0. For such distributions, the useful analog is the probability density function (PDF). The integral of the PDF is the probability over the interval. If the continuous random variable X has PDF f, then

\[\Pr(a < X \leq b) = \int_a^b f(x) dx\]

The relationship between the CDF and the PDF is

\[F(x) \int_{-\infty}^{x} f(t) dt\]

The “Cdf” functions described in this chapter evaluate cumulative distribution functions.

For (absolutely) continuous distributions, the value of F(x) uniquely determines x within the support of the distribution. Thus, the inverse-CDF is a proper inverse function on the interval (0,1) and

\[F^{-1}(F(x)) = x .\]

The “InverseCdf” functions described in this chapter compute the inverses of the distribution functions. The inverses are defined only over the open interval (0,1).

Parameter Notation and Estimation

To emphasize the dependence of a PDF or CDF on one or more parameters, we use the following notation:

\[f (x|θ) \text{ or } F (x|θ)\]

where θ represents one or more distributional parameters. The vertical bar “|” is read as “given”. Some authors prefer to use the semi-colon instead of the vertical bar, as in \(f(x;\theta)\), to emphasize this dependency. However, the vertical bar is more consistent with the notation used for conditional probability distributions. For instance, in the Bayesian framework, there is a distribution on the parameters, so in that sense the probability distributions treated in this chapter are conditional distributions.

A related task to evaluating a probability density or distribution function is to estimate the values of its parameters. For many of the distributions covered in this chapter, function maxLikelihoodEstimates provides maximum likelihood estimates of the unknown parameter values given a sample of observations.

Additional Comments

1. In order to keep the calling sequences simple, whenever possible the functions described in this chapter are written for standard forms of statistical distributions. Hence, the number of parameters for any given distribution may be fewer than the number often associated with the distribution. For example, while a gamma distribution is often characterized by two parameters (or even a third, “location”), there is only one parameter that is necessary, the “shape”.
2. The “scale” parameter can be used to scale the variable to the standard gamma distribution. Also, the functions relating to the normal distribution, normalCdf and normalInverseCdf, are for a normal distribution with mean equal to zero and variance equal to one. For other means and variances, it is very easy to standardize the variables by subtracting the mean and dividing by the square root of the variance.
3. Whenever a probability close to 1.0 results from a call to a distribution function or is to be input to an inverse function, it is often impossible to achieve good accuracy because of the nature of the representation of numeric values. In this case, it may be better to work with the complementary distribution function (one minus the distribution function). If the distribution is symmetric about some point (as the normal distribution, for example) or is reflective about some point (as the beta distribution, for example), the complementary distribution function has a simple relationship with the distribution function. For example, to evaluate the standard normal distribution at 4.0, using normalInverseCdf directly, the result to six places is 0.999968. Only two of those digits are really useful, however. A more useful result may be 1.000000 minus this value, which can be obtained to six significant figures as 3.16713E-05 by evaluating normalInverseCdf at -4.0. For the normal distribution, the two values are related by \(\Phi(x)=1-\Phi(-x)\), where \(\Phi(\cdot)\) is the normal distribution function. Another example is the beta distribution with parameters 2 and 10. This distribution is skewed to the right, so evaluating betaCdf at 0.7, 0.999953 is obtained. A more precise result is obtained by evaluating betaCdf with parameters 10 and 2 at 0.3. This yields 4.72392E-5. (In both of these examples, it is wise not to trust the last digit.)
4. Many of the algorithms used by functions in this chapter are discussed by Abramowitz and Stegun (1964). The algorithms make use of various expansions and recursive relationships and often use different methods in different regions.
5. Cumulative distribution functions are defined for all real arguments, however, if the input to one of the distribution functions in this chapter is outside the range of the random variable, an error of Type 1 is issued, and the output is set to zero or one, as appropriate. A Type 1 error is of lowest severity, a “note”, and, by default, no printing or stopping of the program occurs. The other common errors that occur in the functions of this chapter are Type 2, “alert”, for a function value being set to zero due to underflow, Type 3, “warning”, for considerable loss of accuracy in the result returned, and Type 5, “terminal”, for incorrect and/or inconsistent input, complete loss of accuracy in the result returned, or inability to represent the result (because of overflow). When a Type 5 error occurs, the result is set to NaN (not a number, also used as a missing value code).
6. For distributions with finite ranges, such as the beta distribution, the CDF is 0 for values less than the left endpoint and 1 for values greater than the right endpoint. The functions described in this chapter return the correct values for the distribution functions when values outside of the range of the random variable are input, but warning error conditions are set in these cases.