Introduction

The uniform distribution is a fundamental concept in probability and statistics that describes a scenario where all outcomes within a given range are equally likely to occur. This distribution is characterized by its constant probability density function over the specified interval. Understanding the properties and applications of the uniform distribution is crucial for various fields, including random sampling, quality control, and simulations.

Uniform Distribution Properties

The uniform distribution is defined by two parameters: the lower bound ‘$a$’ and the upper bound ‘$b$’ of the interval $[a, b]$. Its probability density function (PDF) is given by:

$$f(x)=\frac{1}{b-a} \text { for } a \leq x \leq b$$

Here, ‘$a$’ and ‘$b$’ define the range of possible outcomes, and the PDF ensures that the probability is evenly distributed within that range.

Examples: For discrete uniform distributions, consider a fair six-sided die. Each face has an equal chance of landing face up, making it an example of a discrete uniform distribution. For continuous uniform distributions, think of a random point falling within a specified interval on the number line. In quality control, a factory might use a uniform distribution to model the occurrence of defects within a certain range.

Calculating Probabilities

To find the probability of an outcome falling within a certain subinterval, simply divide the width of the subinterval by the width of the entire interval. For instance, if we have a uniform distribution over $[0,10]$ and want to find the probability of landing between 2 and 6, the calculation is:

$$P(2 \leq x \leq 6)=\frac{6-2}{10-0}=\frac{4}{10}=0.4$$

Mean and Variance

The mean $(\mu)$ and variance $(\sigma^2)$ of a uniform distribution are calculated as follows:

• Mean: $\mu=\frac{a+b}{2}$
• Variance: $\sigma^2=\frac{(b-a)^2}{12}$