Uniform Distribution

Probability Distributions

Equal Probability Everywhere — Maximum Ignorance

The uniform distribution assigns equal probability density across an interval. It is the maximum entropy distribution — the most honest choice when you know nothing except the bounds.

Random number generation — all PRNGs start with Uniform(0,1)
Bayesian priors — "I have no prior information" → Uniform
Rounding errors — modeling quantization noise
Simulation — the source distribution for Monte Carlo methods

The uniform is the mother of all distributions — via the inverse transform, it births every other distribution.

Core Concepts

The continuous uniform distribution assigns equal probability density across an interval $[a, b]$ . It is the simplest continuous distribution and serves as the foundational building block from which all other continuous distributions can be derived via the inverse transform method.

DfUniform Distribution

A continuous random variable $X$ has a uniform distribution on $[a, b]$ if its probability density function is constant on $[a, b]$ and zero elsewhere. Written $X \sim \text{Unif}(a, b)$ .

Formally, $X$ is $\text{Unif}(a, b)$ if and only if for any measurable subset $A \subseteq [a, b]$ :

P(X \in A) = \frac{\lambda(A)}{b - a}

where $\lambda(A)$ denotes the Lebesgue measure (length) of $A$ .

PDF of Uniform Distribution

f(x) = \frac{1}{b - a}, \quad a \leq x \leq b

Here,

$a$ =Lower bound of the interval
$b$ =Upper bound of the interval
$1/(b-a)$ =Constant density across [a, b]

ThUniform Distribution is the Maximum Entropy Distribution on [a, b]

Among all continuous distributions supported on $[a, b]$ with finite mean, the uniform distribution maximizes the differential entropy. That is, if $g$ is any other pdf on $[a, b]$ , then:

-\int_a^b f(x) \ln f(x) \, dx \geq -\int_a^b g(x) \ln g(x) \, dx

with equality if and only if $g = f$ .

Proof sketch: The differential entropy of the uniform is $h(X) = \ln(b - a)$ . For any other pdf $g$ on $[a, b]$ , by Gibbs' inequality (or KL divergence $\geq 0$ ):

\int_a^b g(x) \ln \frac{g(x)}{f(x)} \, dx \geq 0

Expanding: $\int g \ln g \, dx - \int g \ln f \, dx \geq 0$ . Since $\ln f = -\ln(b-a)$ is constant:

\int g \ln g \, dx + \ln(b-a) \geq 0 \implies -\int g \ln g \, dx \leq \ln(b-a)

which equals $h(X)$ .

Why It Matters

The uniform distribution is the foundation for random number generation. All pseudo-random number generators produce $\text{Unif}(0,1)$ samples, and the inverse transform method converts these to any desired distribution: if $F$ is a CDF, then $F^{-1}(U) \sim$ that distribution for $U \sim \text{Unif}(0,1)$ .

Derivation of Mean and Variance

Uniform Mean and Variance

E[X] = \frac{a+b}{2}, \quad \text{Var}(X) = \frac{(b-a)^2}{12}

Here,

$a, b$ =Interval endpoints
$(a+b)/2$ =Midpoint of the interval
$(b-a)^2/12$ =Variance depends on interval width

ThDerivation of E[X] and Var(X)

Mean:

E[X] = \int_a^b x \cdot \frac{1}{b-a} \, dx = \frac{1}{b-a} \cdot \frac{x^2}{2} \Big|_a^b = \frac{b^2 - a^2}{2(b-a)} = \frac{(b-a)(b+a)}{2(b-a)} = \frac{a+b}{2}

Second moment:

E[X^2] = \int_a^b x^2 \cdot \frac{1}{b-a} \, dx = \frac{1}{b-a} \cdot \frac{x^3}{3} \Big|_a^b = \frac{b^3 - a^3}{3(b-a)} = \frac{a^2 + ab + b^2}{3}

Variance:

\text{Var}(X) = E[X^2] - (E[X])^2 = \frac{a^2 + ab + b^2}{3} - \frac{(a+b)^2}{4}

= \frac{4(a^2 + ab + b^2) - 3(a^2 + 2ab + b^2)}{12} = \frac{a^2 - 2ab + b^2}{12} = \frac{(b-a)^2}{12}

CDF and Its Properties

CDF of Uniform Distribution

F(x) = \begin{cases} 0 & x < a \\ \frac{x - a}{b - a} & a \leq x \leq b \\ 1 & x > b \end{cases}

Here,

$x$ =Value at which to evaluate CDF
$a, b$ =Interval endpoints

Linearity of the CDF

The CDF $F(x) = (x-a)/(b-a)$ is linear in $x$ . This means: if $X \sim \text{Unif}(a,b)$ , then $P(X \leq c) = (c - a)/(b - a)$ for any $c \in [a, b]$ . For example, the probability that $X$ falls in the left quarter of $[a, b]$ is exactly $0.25$ .

Higher Moments and Moment Generating Function

MGF of Uniform Distribution

M_X(t) = E[e^{tX}] = \frac{e^{tb} - e^{ta}}{t(b - a)}, \quad t \neq 0

Here,

$t$ =Transform variable
$a, b$ =Interval endpoints

ThDerivation of the MGF

M_X(t) = \int_a^b e^{tx} \cdot \frac{1}{b-a} \, dx = \frac{1}{b-a} \cdot \frac{e^{tx}}{t} \Big|_a^b = \frac{e^{tb} - e^{ta}}{t(b-a)}

From this, all moments follow by differentiation: $E[X^n] = M_X^{(n)}(0)$ . For instance:

M_X'(0) = \frac{a+b}{2}, \quad M_X''(0) = \frac{a^2 + ab + b^2}{3}

nth Raw Moment (Closed Form)

E[X^n] = \frac{b^{n+1} - a^{n+1}}{(n+1)(b - a)}

Here,

$n$ =Moment order

The Inverse Transform Method

ThInverse Transform Sampling Theorem

Let $U \sim \text{Unif}(0, 1)$ and let $F$ be any CDF with inverse $F^{-1}$ (the quantile function). Then $X = F^{-1}(U)$ has CDF $F$ .

Proof: $P(X \leq x) = P(F^{-1}(U) \leq x) = P(U \leq F(x)) = F(x)$ , since $U$ is uniform on $[0,1]$ and $F(x) \in [0,1]$ .

Worked Example: Generating Exponential Samples

To generate $X \sim \text{Exp}(\lambda)$ : its CDF is $F(x) = 1 - e^{-\lambda x}$ . Set $u = 1 - e^{-\lambda x}$ and solve:

e^{-\lambda x} = 1 - u \implies x = -\frac{\ln(1-u)}{\lambda}

Since $1 - U \sim \text{Unif}(0,1)$ , we can simplify to $X = -\ln(U)/\lambda$ . This is the standard method used in all statistical software.

Relationship to Other Distributions

The order statistics of $n$ i.i.d. $\text{Unif}(0,1)$ variables follow a Beta distribution: the $k$ -th order statistic is $\text{Beta}(k, n-k+1)$ .
The sum of $n$ i.i.d. $\text{Unif}(0,1)$ variables follows the Irwin-Hall distribution, which approaches $N(n/2, n/12)$ by the CLT.
$\text{Unif}(0,1) = \text{Beta}(1,1)$ .

Specific Applications

Random number generation — The $\text{Unif}(0,1)$ is the source distribution for all Monte Carlo simulations and pseudo-random generators.
Order statistics and ranking — The distribution of sorted uniform samples gives Beta distributions, used in nonparametric statistics.
Bayesian noninformative priors — $\text{Unif}(0,1)$ serves as a prior for probabilities in Bayesian inference (equivalent to $\text{Beta}(1,1)$ ).
Quantization and rounding errors — Modeling the distribution of continuous variables after discrete rounding.

Key Takeaways

Summary: Uniform Distribution

Equal density across $[a, b]$ : $f(x) = 1/(b-a)$
Mean: midpoint $(a+b)/2$ , Variance: $(b-a)^2/12$
CDF is linear: $F(x) = (x-a)/(b-a)$
Maximum entropy distribution on a bounded interval
Foundation for random number generation via inverse transform method
MGF: $M_X(t) = (e^{tb} - e^{ta})/(t(b-a))$
Order statistics of uniforms yield Beta distributions

Uniform Distribution — Equal Probability Across a Range

Uniform Distribution

Equal Probability Everywhere — Maximum Ignorance

Core Concepts

DfUniform Distribution

PDF of Uniform Distribution

ThUniform Distribution is the Maximum Entropy Distribution on [a, b]

Derivation of Mean and Variance

Uniform Mean and Variance

ThDerivation of E[X] and Var(X)

CDF and Its Properties

CDF of Uniform Distribution

Higher Moments and Moment Generating Function

MGF of Uniform Distribution

ThDerivation of the MGF

nth Raw Moment (Closed Form)

The Inverse Transform Method

ThInverse Transform Sampling Theorem

Relationship to Other Distributions

Specific Applications

Key Takeaways

Summary: Uniform Distribution

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