Discrete uniform distribution

Discrete Uniform Distribution is a type of probability distribution that is fundamental to the study of statistics and probability theory. It is a model used to describe the outcome of experiments where each outcome is equally likely to occur. This distribution is discrete because it deals with distinct or separate values.

Definition

The discrete uniform distribution is defined over a finite set of outcomes. If there are n equally likely outcomes, then the probability of each outcome is \( \frac{1 }{ n } \). Mathematically, if a random variable X follows a discrete uniform distribution over n outcomes, the probability mass function (PMF) of X is given by:

\[ P(X = x) = \frac{1}{n} \]

for \( x = 1, 2, 3, ..., n \).

Characteristics

Mean

The mean, or expected value, of a discrete uniform distribution is the average of all possible outcomes. It is calculated as:

\[ \mu = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{n + 1}{2} \]

Variance

The variance, which measures the spread of the distribution, is calculated as:

\[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2 = \frac{n^2 - 1}{12} \]

Applications

The discrete uniform distribution is widely used in various fields such as computer science, for generating random numbers, in game theory for analyzing fair games, and in quality control and survey sampling for making statistical inferences about populations.

Examples

An example of a discrete uniform distribution is the roll of a fair six-sided die. Each side (1 through 6) is equally likely to come up, so the distribution of outcomes is uniform with \( n = 6 \).

Another example is a lottery draw from a well-mixed drum containing numbered balls. If the number of balls is known and each has an equal chance of being drawn, the selection process follows a discrete uniform distribution.

See Also

• Probability theory
• Statistics
• Probability distribution
• Random variable