The Probability Density and Cumulative Distribution Functions

Designations

From probability and statistics, given a continuous random variable X, we denote:

The probability density function, pdf as f(x). (Note: This function is also known as the probability distribution function and the probability mass function, but will be referred to henceforth as the probability density function.)
The cumulative distribution function, cdf, as F(x).

The pdf and cdf give a complete description of the probability distribution of a random variable.

Definitions

If X is a continuous random variable, then the probability density function, pdf, of X, is a function f(x) such that for two numbers, a and b with :

(1)

That is, the probability that X takes on a value in the interval [a, b] is the area under the density function from a to b.

The cumulative distribution function, cdf, is a function F(x) of a random variable, X, and is defined for a number x by:

(2)

That is, for a given value x, F(x) is the probability that the observed value of X will be at most x.

Note that the limits of integration depend on the domain of f(x). For example, for all the distributions considered in this reference, this domain would be [0, +], [-, +] or [γ, +]. In the case of [γ, +] we use the constant γ to denote an arbitrary non-zero point (or a location that indicates the starting point for the distribution). Figure 3-1 illustrates the relationship between the probability density function and the cumulative distribution function.

Figure 3-1: A graphical representation of the relationship between the pdf and cdf.

Mathematical Relationship Between the pdf and cdf

The mathematical relationship between the pdf and cdf is given by:

(3)

Conversely:

In plain English, the value of the cdf at x is the area under the probability density function up to x, if so chosen. It should also be pointed out that the total area under the pdf is always equal to 1, or mathematically:

An example of a probability density function is the well-known normal distribution, whose pdf is given by:

where μ is the mean and σ is the standard deviation. The normal distribution is a two-parameter distribution, i.e. with two parameters μ and σ.

Another two-parameter distribution is the lognormal distribution, whose pdf is given by:

where t' is the natural logarithm of the times-to-failure, μ' is the mean of the natural logarithms of the times-to-failure and σ' is the standard deviation of the natural logarithms of the times-to-failure, t'.