Grade 11 → Probability and Statistics → Probability Distributions ↓

Uniform Distribution

In the world of probability and statistics, the concept of the uniform distribution is one of the simpler and more fundamental probability distributions. The uniform distribution is often considered the most basic form because its main principle is that each outcome is equally likely. The allure of the uniform distribution lies in its simplicity and its wide range of applications in real-life scenarios.

Understanding equitable distribution

Uniform distribution is a type of probability distribution in which all outcomes are equally likely. To understand the topic in depth, let us first look at its main features and see how it works both in theory and practice.

The uniform distribution is essentially described for discrete and continuous random variables:

Discrete uniform distribution: If a finite set of discrete outcomes are equally probable, then the discrete uniform distribution makes sense. A classic example would be a fair dice roll, where each of the six faces has an equal probability of coming up.
Continuous uniform distribution: Here, any value in the continuous range is equally likely. A perfect example is choosing a random number between 0 and 1.

Mathematical representation

Discrete uniform distribution

The probability mass function (PMF) for a discrete uniform distribution is quite simple. If X is a discrete uniform random variable on n outcomes, its PMF is given by:

P(X = x) = 1/n

Where:

X represents a discrete random variable.
x represents the possible outcome.
n is the total number of possible outcomes.

Let us consider throwing a fair six-sided dice. The set of possible outcomes is {1, 2, 3, 4, 5, 6}. The probability of each face is:

P(die = x) = 1/6

Sustainable uniform distribution

For a continuous uniform distribution, the probability density function (PDF) is defined differently. If X is a continuous uniform random variable on the interval [a, b], then its PDF is:

f(x) = 1/(b - a); for a ≤ x ≤ b

and f(x) = 0 for any x not in [a, b].

This graph shows a continuous uniform distribution from a to b, where each point within this interval has an equal probability of occurring.

Properties of uniform distribution

Expectation and variation

Like all probability distributions, the uniform distribution also has characteristics such as expectation (mean) and variance that provide insight into its behavior.

Mean of a discrete uniform distribution:
```
E(X) = (n + 1)/2
```
Variance of the discrete uniform distribution:
```
Var(X) = (n² - 1)/12
```
Mean of a continuous uniform distribution:
```
E(X) = (a + b)/2
```
Variance of a continuous uniform distribution:
```
Var(X) = (b - a)²/12
```

Examples of uniform distribution

Real world example 1: The lottery

Consider a simple lottery where you pick a number between 1 and 50. If the lottery is fair, each number should have an equal chance of being drawn. So, the probability distribution for picking a specific number, say X is 25, is:

P(X = 25) = 1/50

Real-world example 2: Random number generator

Imagine a computer application that randomly picks a number between 0 and 1. This is a typical scenario for a continuous uniform distribution, where the probability of drawing a number is the same throughout the range.

Applications of uniform distribution

Sampling: When generating random samples for experiments or simulations, it is common to use the uniform distribution as a basis.
Monte Carlo methods: Used in computational algorithms that rely on repeated random sampling.
Random number generator: Often implemented based on pseudo-random algorithms that approximate a continuous uniform distribution over a given range.