probability density function

A probability density function (PDF) is a type of function used in the mathematical subfield of probability to describe the relative probability of occurrence of some value that can take on real-number values (uncountable numbers, i.e., including numbers like π that are not integers or rational numbers). An example PDF would be a function giving the probability that a randomly chosen person has some specific height.

More technically, a PDF is a function of a continuous random variable (that can have any real-number value, at least over some interval) that gives the relative likelihood of the variable taking the given value. Such a probability distribution has the quality that the probability of any specific number is zero (to be "exactly six feet tall" is totally improbable, given measurement of sufficient precision), but shows the probability of occurrence within an interval (e.g., being within 1/100 of an inch of six feet tall). This is calculated by integrating the PDF over the interval. If the result of such integration gives the actual probability, the function is termed a normalized PDF (NPDF or N-PDF), in which case, integrating the function over all real numbers (from negative to positive infinity) yields 1. Short of that, a function that gives relative probabilities (e.g., the true ratio of the number of people within 1/100 of an inch of six feet versus similarly for five feet), then it is termed an unnormalized PDF and the equivalent NPDF can be derived by taking the unnormalized PDF and incorporating an additional constant factor (normalizing constant aka normalization constant) consisting of the reciprocal of the area under the entire curve of the unnormalized PDF.

Similarly, a probability mass function (PMF) is like a PDF except that rather than mapping a continuous random variable, it maps a discrete random variable, i.e., a finite (or countable) number of values. An example PMF is the function mapping each possible outcome of a coin flip (heads and tails) to its probability.