Uniform Distribution
The uniform distribution is a continuous probability distribution that describes events where all values within a given range are equally likely. Each of the n possible values has the same probability of occurring (1/n). The distribution is characterized by two parameters: the lower limit (a) and the upper limit (b), which define the range of possible outcomes. The probability density function (PDF) is constant within the interval [a, b] and zero outside this range [1].
Probability Density Function (PDF)
The probability density function (PDF) of a uniform distribution provides the likelihood of any given value within the range \([a, b]\). Mathematically, the PDF is expressed in Equation 1, as defined by Ross [2].
| \[ f(x) = \begin{cases} \frac{1}{b - a}, & \text{if } a \leq x \leq b \\ 0, & \text{otherwise} \end{cases} \] | (1) |
Cumulative Distribution Function (CDF)
The cumulative distribution function (CDF) of the uniform distribution, denoted as F(x), describes the cumulative probability up to a value x. The CDF is particularly useful in determining the probability that a random variable will fall within a specific range [2]. It is defined in Equation 2.
| \[ F(x) = \begin{cases} 0, & \text{if } x < a, \\ \frac{x - a}{b - a}, & \text{if } a \leq x \leq b, \\ 1, & \text{if } x > b. \end{cases} \] | (2) |
Applications in Engineering
Uniform distributions frequently appear in engineering and physical sciences, modeling situations where outcomes are equally likely across a defined range. They are also discussed extensively in the context of probability and statistics, providing a foundational understanding for more advanced distributions [1]. Examples include modeling uncertainties in initial design parameters or in simulations where random sampling from a specific interval is needed [3].