# Uniform Distribution

**Episode #6 of the course Theory of probability by Polina Durneva**

Good morning to you all!

Let’s talk about the uniform distribution today! There actually exist two types of uniform distributions: the discrete one and the continuous one. In order to cover as much material as possible in this course, let’s skip the continuous uniform distribution and talk about the discrete one. Once we finish talking about the continuous distributions (the final two lessons of the course), you can apply similar ideas to the uniform one, but you’ll have to wait a little bit.

**What Is Uniform Distribution?**

The uniform distribution, also known as rectangular distribution, refers to the events with the same probability. The sum of all probabilities from one uniform distribution equals to 1, or 100%. The two most popular and basic examples of the uniform distribution are a fair die and a fair coin. The probability of getting any of 6 sides of a fair die is 1/6 (and their sum equals to 1), and the probability of getting either heads or tails after tossing a fair coin is 1/2.

**Formula for Uniform Distribution**

The formula for the uniform distribution is fairly easy: P(x) = 1/n, where n is the total number of events. For example, if we have 20 different events (n = 20) that follow the uniform distribution, the probability of any of these events to happen is 1/20, or 5%. Once again, this probability distribution is simple and as exciting as other types of distributions.

**Expected Value of Uniform Distribution**

To find the expected value of the discrete uniform distribution, the following formula can be used: (n + 1)/2, where n is the total number of events. The average, or the expected value, of our example with 20 events that follow the uniform distribution would then be: 20 + 1/2 = 10.5. (Please note that the expected value for the continuous uniform distribution is calculated differently with minimums and maximums involved.)

**Variance of Uniform Distribution**

The variance of uniform distribution is slightly more complicated and takes the following form to derive: (n^{2} – 1)/12, where n is the total number of events. If there are 20 uniformly distributed events, the variance of them would be: (20^{2} – 1)/12 = 33.25. (Once again, for the continuous uniform distribution, the variance is calculated in a different way!)

That is it for today’s lesson! I hope you found the discrete uniform distribution interesting, even though it is fairly easy. Tomorrow, we will talk about the geometric distribution that you can use to calculate the probability of getting success at n^{th} trial. This distribution is very different from what you have seen before in this course and should thus be of a great interest to you! After the geometric distribution, we will finish discussing discrete distributions with the negative binomial distribution (which is closely associated with the binomial and geometric distributions).

See you soon,

Polina

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