# Distribution function terminology (PDF, CDF, PMF, etc.) [duplicate]

I am confused about the following terminologies:

1. Distribution Function
2. Cumulative Distribution Function (CDF)
3. Probability Distribution Function
4. Probability Density Function
5. Probability Mass Function (PMF)

(1) Distribution Function == CDF. I.e. they are the same. Am I right?

(2) Which is called PDF: Probability Distribution Function or Probability Density Function?

(3) What is the difference between Probability Distribution Function and Probability Density Function?

(4) Which one is the continuous equivalent of PMF, Probability Distribution Function or Probability Density Function?

Die roll examples could be used for the discrete case and picking a number between 1.5 and 2.5 as an example for the continuous case.

As noted by Wikipedia, probability distribution function is ambiguous term:

A probability distribution function is some function that may be used to define a particular probability distribution. Depending upon which text is consulted, the term may refer to:

• a cumulative distribution function,
• a probability mass function, and/or
• a probability density function.

Cumulative distribution function (CDF) is sometimes shortened as "distribution function", it's

$$F(x) = \Pr(X \le x)$$

the definition is the same for both discrete and continuous random variables. In dice case it's probability that the outcome of your roll will be $x$ or smaller.

Probability density function (PDF) is a continuous equivalent of discrete probability mass function (PMF). Probability mass function is

$$f(x) = \Pr(X = x)$$

In dice case it's probability that the outcome of your roll will be exactly $x$.

Probability mass function has no sense for continuous random variables since $\Pr(X=x)=0$ for continuous random variables (check also Why X=x is impossible for continuous random variables?), because simply a point on real line is so "small" that has no mass and no area.

This leads us to defining probability density as "probability per foot". Simple example is continuous uniform distribution with minimum of $a$ and maximum of $b$, where probability density is the same for each $x$ and equal to

$$f(x) = \frac{1}{b-a}$$

You can easily notice that it changes as the range between $a$ and $b$ (i.e the total area) changes, it is nicely described in Can a probability distribution value exceeding 1 be OK? thread. It is a probability of hitting infinitesimal (infinitely small) interval $[x, x + dx]$ when throwing a dice with infinite number of walls.

• There are fashions here too. For example, it was long customary to insist that probability density functions and probability mass functions were quite different kinds of beasts referring to continuous and discrete variables respectively. A common and in my experience more recent tendency, particularly with mathematically more mature groups, is to insist that the idea of density function is general; it is just a case of density with respect to what kind of measure, and measure could be e.g. counting measure. So density functions, wide sense, include mass functions. – Nick Cox Oct 26 '16 at 13:21

(1) You are right regarding (1).

(2)&(3)&(4) PDF is for probability density function. We usually use probability distribution function to mean CDF. Probability function is used to refer to either probability mass function(the probability function of discrete random variable) or probability density function(the probability function of continuous random variable).

You can also have a look at this What does "probability distribution" mean?