Discrete probability distributions relate to distributions for discrete variables. As described in Random variables, discrete variables are distinct and countable variables, like number of persons, number of visitors, prices on goods, etc.

Discrete probability distributions example:

How many times over the next 100 times that I go to the post office at 2 pm can I expect to have a line with 3 persons in front of me?

This question refers to the number of persons. The number cannot be a number of persons with a long tail of decimals. Persons are not divisible. So, this question relates to a discrete probability distribution.

Based on observations that I would have carried out; I can construct the probability distribution which will help giving an estimate of how many times I can expect there to be 3 persons in line over my next 100 visits in the post office at 2 pm.

Web shop example

Let’s expand on the post-office-example from above:

Say that I am selling goods through a web shop and that I do the shipment via my local post office and that I go there at 2 pm some 100 times a year. Therefore, I am interested in knowing how many people are in line (although, what really interests me, is the amount of time I must wait and not the number of people, but for the sake of the exercise…).

Say I have done 60 observations, so far:

Based on my 60 observations in the post office, there is a 0.13 probability that I will find 3 persons in line. So, I would estimate that over the next 100 visits, for 0.13 x 100 = 13 of these visits there will 3 persons in line.

Coin flipping and probability distributions

What is the probability that we get one tail when flipping two fair coins at a time? The number of tails that we can get can be defined as a random variable. We would say: “Let X be the number of tails we flip” and to answer the question we would calculate for “What is the probability of X=1?”

There are four total possible outcomes for flipping two coins at a time:

• Head, Head
• Head, Tail
• Tail, Tail
• Tail, Head

The probability for each of these outcomes has a 1/4. But we are being asked about the probability of 1 tail to occur.

There are two events that we have one tail, {Head, Tail} and {Tail, Head}, so this gives two fourths = one half or 0.5 probability. We would express the probabilities of the four possible outcomes of the random variables in this way:

By listing the probabilities for each of the possible outcomes of the random variables, we have constructed its’ probability distribution which can be visualized through a histogram:

Examples of discrete probability distributions

In my material, I describe and apply the following discrete probability distributions:

• The Bernoulli distribution
• The binomial distribution
• The Poisson distribution
• The geometric distribution
• The hypergeometric distribution

Learning resources

• Discrete probability distributions on Jbstatistics Youtube playlist
• Jbstatistics video: An Introduction to Discrete Random Variables and Discrete Probability Distributions