Uniform Distribution¶

Formula¶

\[ X\sim \mathrm{Uniform}(a,b),\qquad f(x)=\frac{1}{b-a}\ \text{ for } a\le x\le b \]

Plot¶

fn: 1
xmin: 0
xmax: 1
ymin: 0
ymax: 1.2
height: 280
title: Uniform PDF on [0, 1]

Parameters¶

\(a<b\): interval endpoints
\(x\): value in the interval

What it means¶

All values in the interval \([a,b]\) are equally likely in density terms.

What it's used for¶

Baseline continuous distribution.
Random initialization and simulation.

Key properties¶

Mean \((a+b)/2\), variance \((b-a)^2/12\).
Constant density on the interval.

Common gotchas¶

Continuous uniform has \(P(X=x)=0\) for any exact point.
"Equally likely values" for continuous variables refers to equal-length intervals.

Example¶

\(\mathrm{Uniform}(0,1)\) is commonly used for random sampling in simulations.

How to Compute (Pseudocode)¶

Input: distribution parameters and query values (for PMF/PDF/CDF or moments)
Output: distribution quantities

validate parameters
for each query value x (or count k):
  evaluate the PMF/PDF/CDF formula from the card
optionally compute moments/statistics from known closed forms or by summation/integration
return the requested values

Complexity¶

Time: Typically \(O(q)\) for \(q\) query values once parameters are known (assuming constant-time formula evaluation per query)
Space: \(O(q)\) for output values (or \(O(1)\) for a single query)
Assumptions: Parameter estimation/fitting cost is excluded; numerical special-function evaluation can affect constants for some families