Now that you've been introduced to both discrete (Binomial) and continuous (Normal) distributions, and know how to perform a simple test, let's move on with some more distributions. The uniform distribution is a special one, as there is a discrete as well as a continuous version of this distribution!

You will be able to:

• Define a uniform distribution
• Calculate mean and variance of uniform distribution

The Uniform Distribution describes an event where every possible outcome is equally likely. No single outcome carries any more or less probability of happening than any other possible outcome. The Uniform Distribution can be discrete or continuous.

You've seen an example of a Discrete Uniform Distribution before: rolling a 6-sided dice. This idea can be extended to an $n$-sided dice. No matter how many sides the dice has, with a fair dice, you'd be equally likely to roll every side.

If $a$ is the smallest number and $b$ is the biggest number that can be observed (for a fair dice, $a=1$, $b=6$)

Probability Mass Function:

$$p(x) = \dfrac{1}{(b − a+1)}\text{ when } a \leq x \leq b, \text{and} $$ $$p(x) = 0\text{ when }x \notin ]a, b[ $$

Uniform Distribution Mean:

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

Uniform Distribution Variance:

$$Var(X) = \sqrt{\frac{(b-a)(b-a+2)}{12}}$$

An example of a Continuous Uniform distributed variable would be the waiting time for an elevator that could be on any floor in the building when you call it and can take between 0 and 40 seconds to arrive at your floor. Since the elevator is equally likely to be at any given floor, you can assume every amount of time between 0 and 40 seconds before the elevator arrives (decimals and fractions allowed, to an infinite amount of precision). The formulas for a continuous uniform distribution diverge slightly!

Probability Density Function:

$$f(x) = \dfrac{1}{(b − a)}\text{ when } a \leq x < b, \text{and} $$ $$f(x) = 0\text{ when }x \notin [a, b] $$

Uniform Distribution Mean:

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

Uniform Distribution Variance:

$$Var(X) = \frac{(b - a)^2}{12}$$

NOTE: If you're confused why there is a 12 in the denominator of the formula of the variance for a Uniform Distribution, you're not alone. The short answer is that it involves calculus. As a data scientist, you don't need to understand the derivation of this formula and where this 12 comes from. However, if you're interested, this Quora answer gives an excellent explanation!

In this lesson, you learned about the uniform distribution, its applications, and the equations used to compute the mean and variance of a uniformly distributed random variable.