Continuous Uniform Distribution

Compute PDF, CDF, quantiles, moments, and visualize the distribution. The fundamental "rectangular" distribution for equal probability over an interval.

Uniform Distribution X ~ U(a, b):

PDF: f(x) = 1/(b-a), a ≤ x ≤ b

CDF: F(x) = (x-a)/(b-a), a ≤ x ≤ b

Quantile: Q(p) = a + p(b-a), 0 ≤ p ≤ 1

Real number, a < b
Real number, b > a
Enter a real number
? Standard U(0,1)
⚖️ Symmetric U(-1,1)
? U(2,5)
? U(0,10)
? U(100,200)
Continuous Uniform Results
PDF at x = 0.5000 = 1.00000000
f(0.5) = 1/(1-0) = 1.0000
0.5000
Mean (μ)
0.083333
Variance (σ²)
0.288675
Std Dev (σ)
0.5000
Median
PDF f(x)
CDF F(x)
Selected x / quantile

* PDF is constant on [a,b]; CDF increases linearly.

Understanding the Continuous Uniform Distribution

The continuous uniform distribution (also called rectangular distribution) assigns equal probability density to every point in the interval [a, b]. It is the maximum entropy distribution for a given interval and serves as the fundamental model for complete ignorance.

Key Properties
  • PDF: f(x) = 1/(b-a), a ≤ x ≤ b
  • CDF: F(x) = (x-a)/(b-a), a ≤ x ≤ b
  • Quantile: Q(p) = a + p(b-a)
  • Mean: (a+b)/2
  • Variance: (b-a)²/12
  • Skewness: 0 (symmetric)
  • Kurtosis excess: -6/5 (platykurtic)
  • MGF: (e^{tb} - e^{ta})/(t(b-a))
  • Entropy: ln(b-a) nats
Applications
  • Random number generation: basis for simulating other distributions (inverse transform)
  • Bayesian statistics: non‑informative prior for location parameters
  • Rounding errors: uniform distribution of fractional parts
  • Sensitivity analysis: equal probability over parameter ranges
  • P‑values: under null hypothesis, p‑values are U(0,1)
  • Waiting times: arrivals completely random (Poisson process) → interarrival uniform conditional on one event?

Connection to other distributions:

  • If X ~ U(0,1), then Y = -ln(1-X)/λ ~ Exponential(λ).
  • If X₁,...,Xₙ i.i.d. U(0,1), the order statistics follow Beta distributions: X_{(k)} ~ Beta(k, n-k+1).
  • Sum of independent uniforms → Irwin–Hall distribution (approaches normal).
  • The uniform is the maximum entropy distribution given only bounds.

Frequently Asked Questions (6 items)

Discrete uniform assigns equal probability to a finite set of values (e.g., dice roll). Continuous uniform assigns equal probability density over an interval — probability of any exact point is zero, only intervals have non‑zero probability.

For a ≤ c ≤ d ≤ b, P(c ≤ X ≤ d) = (d-c)/(b-a). This calculator gives CDF at a point; subtract CDF values to get interval probability.

Derived from Var = E[X²] - (E[X])². For uniform, E[X²] = ∫ x²/(b-a) dx = (b³-a³)/(3(b-a)) = (a²+ab+b²)/3. Subtract ((a+b)/2)² = (a²+2ab+b²)/4, combine to get (b-a)²/12.

No — the interval [a,b] requires a < b. The calculator will swap values if a > b to ensure a ≤ b, with a warning.

Most computer random number generators produce U(0,1) pseudo‑random numbers. By applying the inverse CDF (quantile function), samples from any continuous distribution can be generated.

Given i.i.d. sample, the MLEs are â = min(Xᵢ), b̂ = max(Xᵢ). They are biased but consistent; unbiased estimators are â = (n·min - max)/(n-1)? Actually for uniform, the minimal sufficient statistics are min and max.