The Probability Mass Function

- and how you can use it to calculate probabilities

Probability distributions describe how probabilities are distributed over the possible values that a random variable can take. A random variable can be discrete- taking on only certain fixed values such as 0,1,2,… etc., or it can be continuous- taking any numerical values in an interval or collection of intervals.

In this article, I will focus on some frequently used discrete distributions, and attempt to explain the mathematical formulation of their Probability Mass Function, using simple examples.

The Probability Mass Function, or the PMF, provides the probability for each value of the random variable.

It is denoted by fₓ(x), where X is the random variable. Let us say that random variable X, takes an arbitrary value k. The PMF will give us the probability of that happening. Mathematically,

(i) Positivity

A probability cannot be negative, and since fₓ(x) is a probability, it has to greater than or equal to zero.

(ii) Normalization

The sum of fₓ(x) over all possible outcomes k of the random variable is 1. This expression gives the probability of random variable X taking on any value, which is 1.

This concludes the not so interesting definition part! We are now equipped with the knowledge that we need to delve into the various distributions.

The random variable X has a binary outcome. Let us denote these outcomes by 1 and 0. If the probability of X being 1 is p, then the probability of X being 0 is 1-p (Normalization property!).

A simple example of this is the tossing of a fair coin i.e there is a 50% chance of it landing heads up and a 50% chance of it landing tails up. Let 1 denote heads and 0 denote tails. Then the PMF can be described as-

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