Uniform Distribution
Print PageA Uniform Distribution Calculator (also known as a Continuous Rectangular Distribution Calculator, Flat Probability Density Utility, Uniform CDF & PDF Generator, or Discrete Uniform Probability Analyzer) computes exact Probability Density Function height values (f(x) = 1 ÷ [b - a] for a ≤ x ≤ b), Cumulative Distribution Function probabilities (F(x) = [x - a] ÷ [b - a]), sub-interval probabilities (P(x1 ≤ X ≤ x2) = [x2 - x1] ÷ [b - a]), expected arithmetic mean (μ = [a + b] ÷ 2), median (x0.5 = [a + b] ÷ 2), variance (Var[X] = (b - a)2 ÷ 12), and standard deviation (σ = (b - a) ÷ √12 ≈ 0.288675 · [b - a]).
In public transit waiting time modeling, computer pseudo-random number generation (PRNG seeds), industrial manufacturing tolerance checks, and Bayesian non-informative prior modeling, the uniform (rectangular) distribution represents scenarios where every equal-length sub-interval shares an identical probability of occurrence.
Our free online Uniform Distribution Calculator provides instant calculations across all continuous and discrete flat probability parameters:
- Continuous Probability Density Function Height (PDF for a ≤ x ≤ b):
f(x; a, b) = 1 ÷ ( b - a )(Zero outside[a, b]). - Cumulative Distribution Function (CDF / P[X ≤ x]):
F(x; a, b) = ( x - a ) ÷ ( b - a ). - Sub-Interval Probability Formula (P[x1 ≤ X ≤ x2]):
P(x1 ≤ X ≤ x2) = ( x2 - x1 ) ÷ ( b - a ). - Expected Arithmetic Mean (μ) & Median (x0.5):
μ = Median = ( a + b ) ÷ 2. - Variance (Var[X]) & Standard Deviation (σ):
Var[X] = ( b - a )2 ÷ 12 &implies; σ = ( b - a ) ÷ √12. - Discrete Uniform Distribution (N Equiprobable Outcomes):
P(X = k) = 1 ÷ NandVar[X] = [ (b - a + 1)2 - 1 ] ÷ 12.
Master Uniform Reference Table (City Bus Arrival Waiting Time: a = 0 min, b = 15 min)
The table below displays waiting time boundaries (x min), exact PDF density heights f(x), lower cumulative probabilities P(X ≤ x), upper tail probabilities P(X > x), and expected values for a bus arriving uniformly between 0 and 15 minutes (a = 0.0 min, b = 15.0 min, b – a = 15.0 min):
| Waiting Time Threshold (x Minutes) | Constant Density Height f(x) | Lower Cumulative CDF P(X ≤ x) | Upper Tail Probability P(X > x) | Transit Passenger Experience Status |
|---|---|---|---|---|
| 3.0 minutes | 1 ÷ 15 = 0.06667 (6.67%/min) | 20.0000% (0.2000) | 80.0000% | 20.0% Fast Arrival Chance (≤3 min) |
| 5.0 minutes | 0.06667 (6.67%/min) | 33.3333% (0.3333) | 66.6667% | 1/3 Probability of Arrival Within 5 min |
| 7.5 minutes (Mean μ & Median) | 0.06667 (6.67%/min) | 50.0000% (0.5000) | 50.0000% (0.5000) | Expected Average Wait (μ = 7.50 min) |
| 10.0 minutes | 0.06667 (6.67%/min) | 66.6667% (0.6667) | 33.3333% (33.33%) | 33.3% Delayed Tail Wait (>10 min) |
| 12.0 minutes | 0.06667 (6.67%/min) | 80.0000% (0.8000) | 20.0000% (20.00%) | 80% Probability Arrived Before 12 min |
Step-by-Step Bus Arrival Waiting Time Calculation (0 to 15 Minutes)
To evaluate passenger waiting times for a city bus arriving uniformly between a = 0.0 minutes and b = 15.0 minutes:
Step 1 (Calculate PDF Density Height f(x)): f(x) = 1 ÷ (15.0 - 0.0) = 1 ÷ 15.0 = 0.06667 (6.67% probability per minute)
Step 2 (Calculate Expected Mean & Median Wait μ): μ = (0.0 + 15.0) ÷ 2 = 15.0 ÷ 2 = 7.50 minutes
Step 3 (Calculate Variance & Standard Deviation σ): Var[X] = (15.0)^2 ÷ 12 = 225.0 ÷ 12 = 18.75 &implies; σ = √18.75 = 4.330 minutes
Step 4 (Calculate Probability of Waiting Less Than 5.0 Minutes P[X ≤ 5.0]): F(5.0) = (5.0 - 0.0) ÷ 15.0 = 5.0 ÷ 15.0 = 0.3333 ≈ 33.33%
Step 5 (Calculate Probability of Waiting Between 3.0 and 9.0 Minutes P[3 ≤ X ≤ 9]): P(3 ≤ X ≤ 9) = (9.0 - 3.0) ÷ 15.0 = 6.0 ÷ 15.0 = 0.4000 ≈ 40.00%
Thus, a passenger expects to wait 7.50 minutes on average, with a 33.33% chance of waiting under 5 minutes and a 40.00% chance of waiting between 3 and 9 minutes.
Probability Distributions Comparison: Continuous Uniform vs. Discrete Uniform vs. Normal vs. Exponential
Below is a comparative reference chart detailing when to use the Uniform distribution versus related continuous models:
| Probability Model | Density / Mass Curve Shape | Domain Boundaries | Primary Practical Application |
|---|---|---|---|
| Continuous Uniform Distribution | Flat Constant Rectangular Height f(x) = 1/(b-a) | Bounded [a, b] interval | Scheduled bus waiting times, PRNG random numbers. |
| Discrete Uniform Distribution | Equal Integer Probabilities P(X=k) = 1/N | Finite integer set {1, 2… N} | Fair 6-sided dice rolls, lottery ticket draws. |
| Exponential Distribution | Decaying Exponential Curve f(x) = λ e-λx | Unbounded Positive [0, +&infty;) | Random un-scheduled arrival inter-waiting times. |
| Gaussian (Normal) Distribution | Symmetric Bell Curve centered at μ | All Real Numbers (-&infty;, +&infty;) | Natural measurement errors & standardized test scores. |
History & Mathematics: 1657 Christiaan Huygens to 1812 Pierre-Simon Laplace
1657 Christiaan Huygens & Games of Chance
In 1657, Dutch polymath Christiaan Huygens published De Ratiociniis in Ludo Aleae (On Reckoning in Games of Chance), introducing discrete uniform probabilities for fair dice rolls and card draws.
1763 Thomas Bayes & 1812 Pierre-Simon Laplace
In 1763, Thomas Bayes and later Pierre-Simon Laplace in 1812 established continuous uniform distributions as uninformative non-informative prior distributions under the Principle of Insufficient Reason.
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Frequently Asked Questions (FAQ)
What is the formula for Continuous Uniform Distribution CDF?
The Cumulative Distribution Function is F(x) = ( x - a ) ÷ ( b - a ) for a ≤ x ≤ b.
What is the formula for Uniform Distribution Variance?
The variance of a continuous uniform distribution on interval [a, b] is Var[X] = ( b - a )2 ÷ 12.
What is the difference between Continuous and Discrete Uniform distributions?
A Continuous Uniform distribution applies to real-numbered intervals where any decimal value is possible, whereas a Discrete Uniform distribution applies to a finite set of distinct integer outcomes (such as a 6-sided die).