# Discrete Random Variables: The discrete uniform distribution

## Discrete Random Variables

• A discrete random variable can only take on a finite or countable number of values.
• Each value has an associated probability, defining how likely that outcome is to occur.

## Discrete Uniform Distribution

• The discrete uniform distribution is a type of probability distribution in which all outcomes are equally likely.
• A simple and common example of a discrete uniform distribution is a fair six-sided die. Each face (outcome) is equally likely to occur on each throw.

## Characteristics of a Discrete Uniform Distribution

• All outcomes are equally likely; the probability of any specific outcome is 1/n, where n is the number of outcomes.
• A discrete uniform distribution is defined by the smallest and largest values in the range of possible outcomes.
• The outcomes must be finite and countable, and each outcome should be independent of the others.

## Parameters of the Discrete Uniform Distribution

• A discrete uniform distribution is typically denoted as U(a, b) or DU(a, b), where a and b are the smallest and largest possible outcomes respectively.
• The expected value (mean) of a discrete uniform distribution is (a + b)/2.
• The variance is (b - a + 1)² - 1/12, and the standard deviation is the square root of the variance.

## Probability Mass Function (PMF)

• For a discrete uniform distribution, the probability mass function assigns an equal probability 1/n to each outcome.
• With n outcomes, the PMF is P(X = k) = 1/n for a ≤ k ≤ b. ‘k’ is a specific outcome in the distribution’s range.

## Cumulative Distribution Function (CDF)

• For a discrete random variable, the cumulative distribution function gives the probability that the variable can take a value less than or equal to a certain value.
• For a discrete uniform distribution, the CDF is F(k) = (k - a + 1) / (b - a + 1) for a ≤ k ≤ b.

## Application of Discrete Uniform Distribution

• Discrete Uniform Distribution is used in scenarios where we have a finite number of outcomes and each of these outcomes is equally likely to occur.