Probability Mass Function (PMF)

Also known as the Probability Distribution Function or Discrete Probability Function.

This returns the probability of getting "equal to" a particular random variable.

The word "mass" is used because every value has a specific probability.

The convention is to use a lowercase f to represent this function.
This function always returns a value between 0 and 1.

Lets assume that you have the following table of data.
This table shows the age of people within a small company.
The data in this table can be quickly summarised by plotting a frequency distribution graph.
This chart displays vertical lines (or bars) where the height of the line (or bar) represents the frequency.

Discrete Random Variables

A discrete random variable always has an associated probability distribution.

This is a list of probabilities associated with each of the possible values

If you divide the frequency by the total number of occurrences

Suppose we flip a coin and count the number of heads. The number of heads could be any integer value between 0 and plus infinity. However, it could not be any number between 0 and plus infinity. We could not, for example, get 2.5 heads. Therefore, the number of heads must be a discrete variable.

More formally, the probability distribution of a discrete random variable X is a function which gives the probability p(xi) that the random variable equals xi, for each value xi:
p(xi) = P(X=xi)