The discrete uniform distribution is a probability distribution that assigns equal probability to each possible value within a set of mutually exclusive and exhaustive outcomes. In other words, every possible outcome has the same probability of occurring.

For example, let’s say we have a fair six-sided die. Each side of the die is labeled with a number from 1 to 6. When the die is rolled, each number has an equal probability of being the outcome, or 1/6. This is an example of the discrete uniform distribution.

Another example of the discrete uniform distribution is a lottery with 10 possible numbers. In this case, each number has a probability of 1/10 of being drawn.

In both of these examples, the possible outcomes are discrete (meaning they are individual and separate), and the probabilities are uniform (meaning they are the same for each outcome).

The discrete uniform distribution can be represented by a probability mass function, which is a mathematical function that describes the probability of each possible outcome. For the fair six-sided die, the probability mass function would be:

f(x) = 1/6, for x = 1, 2, 3, 4, 5, 6

This function tells us that the probability of each number (x) on the die is 1/6.

Similarly, for the lottery with 10 possible numbers, the probability mass function would be:

f(x) = 1/10, for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

The discrete uniform distribution has some important properties that make it useful in many applications. For instance, the expected value (or mean) of a discrete uniform distribution is equal to the average of the possible outcomes. In the case of the fair six-sided die, the expected value would be 3.5, which is the average of the numbers 1, 2, 3, 4, 5, and 6.

Additionally, the variance of a discrete uniform distribution is equal to the square of the range of possible outcomes divided by 12. In the case of the fair six-sided die, the variance would be 2.92, which is calculated as (6-1)^2 / 12.

The discrete uniform distribution has many practical applications. For example, it can be used to model the outcomes of random events, such as the roll of a die or the draw of a lottery number. It can also be used to model the distribution of certain types of data, such as the distribution of test scores or the distribution of the number of days it takes for a customer to return a product.

In summary, the discrete uniform distribution is a probability distribution that assigns equal probability to each possible value within a set of mutually exclusive and exhaustive outcomes. It is represented by a probability mass function and has useful properties, such as the expected value and variance. It has many practical applications in modeling random events and data distributions.