Does Wolfram Mathworld make a mistake describing a discrete probability distribution with a probability density function?

Usually a probability distribution over discrete variables is described using a probability mass function (PMF):

When working with continuous random variables, we describe probability distributions using a probability density function (PDF) rather than a probability mass function.

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However, Wolfram Mathworld is using PDF to describe the probability distribution over discrete variables:

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Is this a mistake? or it does not much matter?

Answer

It is not a mistake

In the formal treatment of probability, via measure theory, a probability density function is a derivative of the probability measure of interest, taken with respect to a “dominating measure” (also called a “reference measure”). For discrete distributions over the integers, the probability mass function is a density function with respect to counting measure. Since a probability mass function is a particular type of probability density function, you will sometimes find references like this that refer to it as a density function, and they are not wrong to refer to it this way.

In ordinary discourse on probability and statistics, one often avoids this terminology, and draws a distinction between “mass functions” (for discrete random variables) and “density functions” (for continuous random variables), in order to distinguish discrete and continuous distributions. In other contexts, where one is stating holistic aspects of probability, it is often better to ignore the distinction and refer to both as “density functions”.

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Source : Link , Question Author : czlsws , Answer Author : Ben

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