Types of Distributions - Uniform, Bernoulli, and Binomial

Types of Distributions

A few words before we start:

📌 We will be looking at the mathematical representations for the various distributions. You do NOT need to remember them by heart! We are going to look at them so that we can get a complete picture.

📌 The important thing is to be able to identify distributions from their graphs and to know their major properties and/or distinguishing features. For example, say you plot the distribution of an interesting variable in your dataset; you should be able to tell what kind of distribution that variable is following — is it Normal or Poisson or something else?

📌 Pay special attention to the Normal Distribution and its properties; you should know that one really well as you are likely to encounter it most frequently.

1. Uniform Distribution

This is a basic probability distribution where all the values have the same probability of occurrence within a specified range; all the values outside that range have probability of 0. For example, when we roll a fair die, the outcomes can only be from 1 to 6 and they all have the same probability, 1/6. The probability of getting anything outside this range is 0 — you can’t get a 7.

The graph of a uniform distribution curve looks like this:

We can see that the shape of the uniform distribution curve is rectangular. This is the reason why this is often called the rectangular distribution.

The density function, f(X), of a variable X that is uniformly distributed can be written as:

$$f(x)= \frac{1}{b - a}$$

where a and b are the minimum and maximum values of the possible range for X.

The mean and variance of the variable X can then be calculated like so:

$$Mean = E(X) = \frac{a + b}{2}$$ ...