1 What is the Uniform Distribution?
The uniform distribution is a continuous probability distribution where all outcomes are equally likely within a specified interval. It is the simplest continuous distribution and serves as a fundamental model for random variables with no preference for any particular value within a bounded range.
A random variable \(X\) follows a uniform distribution on the interval \([a, b]\) with \(a < b\) if \[ X \sim \text{Uniform}(a, b) \quad \text{or} \quad X \sim U(a,b). \]
The most common case is the standard uniform distribution on \([0,1]\), denoted \(\text{Uniform}(0,1)\) or \(U(0,1)\).
2 Probability Density Function
The probability density function is \[ f(x) = \begin{cases}\frac{1}{b-a} & x \in [a,b] \\ 0 & \text{otherwise}\end{cases} \]
The constant value \(\frac{1}{b-a}\) ensures that the PDF integrates to 1 over the interval \([a,b]\).
The cumulative distribution function is \[ F(x) = \begin{cases}0 & x < a \\ \frac{x-a}{b-a} & a \leq x \leq b \\ 1 & x > b\end{cases} \]
3 Key Properties
\[ \mathbb{E}[X] = \frac{a+b}{2} \] \[ \operatorname{Var}(X) = \frac{(b-a)^2}{12} \]
Proof: \[ \mathbb{E}[X] = \int_a^b x \cdot \frac{1}{b-a} dx = \frac{1}{b-a} \left[\frac{x^2}{2}\right]_a^b = \frac{1}{b-a} \cdot \frac{b^2 - a^2}{2} = \frac{a+b}{2}. \]
For the variance, \(\mathbb{E}[X^2] = \int_a^b x^2 \cdot \frac{1}{b-a} dx = \frac{a^2 + ab + b^2}{3}\), so \[ \operatorname{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2 = \frac{a^2 + ab + b^2}{3} - \frac{(a+b)^2}{4} = \frac{(b-a)^2}{12}. \] \(\square\)
\[ M_X(t) = \frac{e^{tb} - e^{ta}}{t(b-a)}, \quad t \neq 0 \]
For \(t = 0\): \(M_X(0) = 1\).
\[ \varphi_X(t) = \frac{e^{itb} - e^{ita}}{it(b-a)}, \quad t \neq 0 \]
For \(t = 0\): \(\varphi_X(0) = 1\).
4 Affine Transformations
If \(X \sim \text{Uniform}(a, b)\) and \(Y = cX + d\) for constants \(c > 0\) and \(d\), then \[ Y \sim \text{Uniform}(ca + d, cb + d). \]
In particular, if \(X \sim \text{Uniform}(0,1)\), then \(Y = (b-a)X + a \sim \text{Uniform}(a,b)\).
5 Relationship to Order Statistics
The uniform distribution arises naturally through order statistics:
If \(U_1, U_2, \ldots, U_n\) are i.i.d. \(\text{Uniform}(0,1)\), and \(U_{(1)} \leq U_{(2)} \leq \cdots \leq U_{(n)}\) are the order statistics, then the \(k\)-th order statistic follows a Beta distribution: \[ U_{(k)} \sim \text{Beta}(k, n+1-k). \]
6 Examples and Applications
- Random number generation: Uniform(0,1) is the foundation for generating random variables from any other distribution via inverse transform sampling.
- Simulation and modeling: Modeling scenarios where any value in a range is equally likely (e.g., arrival times within a time window).
- Uncertainty quantification: Prior distribution in Bayesian analysis when there is no reason to favor any particular value in a bounded interval.
- Quality control: Acceptable measurements or dimensions within specified tolerances.
- System performance: Response time of a task uniformly distributed between minimum and maximum latency values.