Discrete Distributions & Binary Data
statistics
Binomial, Geometric, Negative Binomial, Hypergeometric, and Poisson.
Introduction
What is a probability disctribution?
A discrete probability distribution is a probability distribution that describes the likelihood of outcomes for a discrete random variable. In other words, it gives the possible values that the variable can take, and the probability associated with each of those values.
What is a discrete variable?
A discrete variable is a variable which can only take a finite, known number of values. This variable is not measured on a continuous scale (e.g. speed, temperature, height), but rather is categorical in nature (e.g. gender, race, rating, class).
Discrete vs. Continuous Variable
Why do we make a distinction between discrete and continuous variables when working with probability distributions? Because the maths used to describe each is different.
The distribution for a discrete variable is created using a Probability Mass Function (PMF). For a continuous variable, the distribution is created using a Probability Density Function(PDF).
The PMF is used to calculate probabilities for a known and finite number of variable outcomes - think Heads vs Tails for a coin flip. There are no values between Heads and Tails.
On the other hand, a PDF can be used to, theoretically, calculate the probability for an infinite number of variable outcomes (think temperature, where there is an infinite number of fractional temperatures between two corresponding integer temperatures).
Know your data.
If you don’t understand your data, you run the risk of using the wrong instrument for the application. You risk obtaining false results and coming to false conclusions.
Definitions
Here are a few definitions to solidify our understanding:
Probability Distribution: A statistical instrument that allows us to determine probabilities and likelihoods for a random variable.
There are various discrete distributions used to answer different classes of questions. We’ll talk about these:
- Binomial
- Poisson
- Geometric
- Negative Binomial
- Hypergeometric
Random Variable: A variable for which we don’t know the next value. Think of drawing a card from a standard deck. This variable is random. We don’t know whether we’ll draw a red or a black, a spade or a heart, a number card or a face card.
Discrete Variable: A variable which can only take finite, known number of values. This variable is not measured on a continuous scale (e.g. speed, temperature, height), but rather is categorical in nature (e.g. gender, race, rating, class).
Binary Variable: A special case of discrete variable - a variable which can only take 1 of 2 values.
- A coin (heads or tails)
- A die (even or odd)
- A rocket launch (go or no-go)
- A scoring attempt (made or missed)
Importance & Application
Basic Discrete Probability Distributions
Bernoulli Distribution
When measuring a binary random variable in a probability distribution, we usually speak in terms of a Bernoulli trial. A Bernoulli trial is a random experiment with only 2 possible outcomes, success or failure. The probabilities associated with a success or failure must be fixed across all trials.
There are different types of distributions used to describe different types of random variables and different types of probabilities. We’ll discuss several distributions:
Bernoulli Trials
To make things more concrete, let’s look at a probability distribution for a single Bernoulli trial - a coin flip (remember, a Bernoulli trial is an instance of a Binary random variable - it can only take 1 of 2 possible values, and it consists of a single trial). A fair coin has a 50% chance of landing on heads (we’ll call this a success), and 50% chance of landing on tails.
As we can see, we have a 50% chance (0.5) of getting 0 successes, and a 50% chance (0.5) of getting 1 success.
We also used a Binomial Distribution to visualize this probability distribution.
Let’s take this a step further. What does a probability distribution look like for 10 coin flips. How many successes (heads) are we most likely to get after 10 experiments?
Since it’s a fair coin, with a 50% chance of success, we’re most likely to see 5 out of 10 successes.
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