# The discrete uniform distribution

## Understanding The Discrete Uniform Distribution

• The Discrete Uniform Distribution is a type of probability distribution in which all outcomes are equally likely.
• This distribution is applicable in situations where each outcome has an equal chance of occurring, such as throwing a fair die or drawing a card from a well-shuffled deck.
• It is denoted as U(n) where n is the total number of outcomes.

## Properties of The Discrete Uniform Distribution

• The discrete uniform distribution has one parameter: n (the number of outcomes)
• The possible values of a discrete uniform random variable can range from 1 to n.
• The mean of a discrete uniform distribution U(n) is (n+1)/2, and the variance is (n^2-1)/12.

## Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)

• The Probability Mass Function of a discrete uniform distribution gives the probability of each outcome. Each outcome is equally likely, so the PMF is 1/n for each outcome.
• The Cumulative Distribution Function for a discrete uniform distribution is the probability that the random variable is less than or equal to a certain value. It can be calculated as x/n for a random variable x.

## Working with The Discrete Uniform Distribution

• The discrete uniform distribution is often used in statistical simulations and modelling where each outcome has an equal chance of happening.
• For example, it can be used to calculate probabilities in games of chance, such as “what’s the probability of drawing a red card from a well-shuffled deck?”
• Understanding the properties and functions of the discrete uniform distribution is essential for working with data that falls under this category.

Remember, better understanding comes when you apply these concepts in various problem scenarios. So be sure to work through your practice problems and exercises.