What's the difference between a probability mass function and a discrete probability distribution?

Question

The Wikipedia page on "probability mass function" says "The probability mass function is often the primary means of defining a discrete probability distribution" (emphasis added). I was under the impression that those two terms were completely synonymous. What's the difference between them?

Answer 1

The difference is subtle. I think it helps to realize that a (probability) distribution is a more general concept. A distribution is something intrinsic to a random variable and determines/describes the behaviour of it. In contrast, a probability mass function (PMF) is one way to concretely describe the distribution quantitatively. Things to note:

• In general, a distribution does not necessarily have a PMF or a probabilty density function (PDF). However, a discrete distribution always has a PMF, and a PMF always describes a discrete distribution, and therefore the two things are basically the same.
• You should think of a PMF as just one way to describe/define a dicrete distribution. We could also use the cummulative density function for example, or any other exotic description we can think of.
• Maybe you can very loosely think like this: The distribution is what carries out the random experiments, and it might look at the PMF to decide what to do, or it might look at something else.

Answer 2

The probability mass function is a function with whom you can calculate probability of a random variable.

Discrete probabilty distribution is about a way a random variable can behave in terms of the values it can assume.

Answer 3

Note that the statement is the probability mass function can define the distribution of a discrete random variable, but the distribution is not the same thing as the distribution function

The probability mass function for a discrete random variable $X$ is $P(X = k)$. The probability distribution function often refers to the cumulative distribution function, $P(X \leq k)$.

I think when speaking or writing, people often drop the 'function' at the end, hopefully it'll be clear from context what they mean.