Probability distribution vs. probability mass function / Probability density function terms: what's the difference

Question

I struggle to understand what's the difference between Probability distribution (https://en.wikipedia.org/wiki/Probability_distribution) vs. probability mass function (https://en.wikipedia.org/wiki/Probability_mass_function) or Probability density function (https://en.wikipedia.org/wiki/Probability_density_function). Both probability distribution and lets say PMF seem to reflect probability of values of a random variable. Note that I do not ask the difference between PDF and PMF.

Consider the following example when a 4-sides dice is rolled twice. X is the sum of two throws. I calculate the probability mass function (left) and then show the result graphically (right). But it seems that it is fair to call this graph probability distribution. Isn't it?

Thanks!

Answer 1

Probability density functions are always associated with continuous random variables. Continuous variables, are variables which can be measured, such as time, and length. However, a probability mass function is a function which only takes in discrete values for its random variable. However; in both cases the function must satisfy two conditions in order to be a PDF or PMF: 1) The honesty condition (The sum of all the values or outcomes must equal one for discrete cases, and integral for continuous cases). 2) Given any outcome, x, the function f(x) must be between 0 and 1. A probability distribution is a function which assigns a probability to an outcome.

Answer 2

"Probability distribution" is a general term describing a mathematical entity that is represented by the "cumulative distribution function" (or just "distribution function") and also by its "probability mass function" or "probability density function" (or just "density"), when it exists.

For example the following sentence is perfectly correct even though a bit wordy: "the cumulative distribution function of the Normal probability distribution is XXX, while its probability density function is YYY".

As to what your graph reflects, the cumulative distribution function is non-decreasing by definition.