Uniform Distribution Calculator

About the Uniform Distribution Calculator

The uniform distribution calculator handles both continuous and discrete uniform distributions. In the continuous case, every value in an interval is equally likely in the sense of density; in the discrete case, every integer in the range has the same probability.

Enter bounds and query values to compute interval probabilities, review the PDF or PMF, and inspect the quantile and order-statistic tables. For continuous distributions, the page also shows how the sampling distribution of the mean tightens as sample size grows.

The result is a compact way to work with the simplest "flat" probability model and to see how it behaves in both theory and simulation.

When This Page Helps

Uniform models are a good baseline because the mathematics is simple and the assumptions are explicit. They are useful whenever you know the range but do not have a reason to prefer one value over another.

This page gives you the core calculations in one place so you can move from the interval assumption to probabilities, quantiles, and sample-size effects without rebuilding the formulas manually.

How to Use the Inputs

1. Select continuous or discrete uniform distribution.
2. Enter the lower bound (a) and upper bound (b).
3. Enter query values x₁ and x₂ to find P(x₁ ≤ X ≤ x₂).
4. For continuous distributions, set sample size to see the sampling distribution.
5. Use presets for common scenarios like die rolls or bus wait times.
6. Review the quantile table and order statistics for detailed analysis.

Example Calculation

Tips & Best Practices

• The uniform distribution has the highest entropy (maximum uncertainty) among distributions with a fixed range.
• For continuous uniform, P(X = exact value) is always 0 — only intervals have nonzero probability.
• The sum of uniform random variables approaches a normal distribution (central limit theorem) — even 3 uniforms summed look quite bell-shaped.
• Discrete uniform is the model for fair dice — a d6 is Discrete Uniform(1, 6) with PMF = 1/6 each.
• Order statistics from uniform distributions have elegant formulas: E[X₍ₖ₎] = a + k(b−a)/(n+1).
• Random number generators produce Uniform(0, 1) values, then transform them to other distributions using inverse CDF methods.

Uniform Distribution as Maximum Entropy Prior

In Bayesian statistics, when you have no prior information about a parameter except its range, the uniform distribution is the maximum entropy prior — it encodes "I know nothing about which values are more likely." This makes it the default choice for uninformative priors in Bayesian analysis.

Inverse Transform Sampling

Every probability distribution can be generated from uniform random numbers. If U ~ Uniform(0,1), then F⁻¹(U) follows the distribution F. This inverse transform method is the basis of all Monte Carlo simulation. For example, −ln(U)/λ generates exponential random variables.

The Irwin-Hall Distribution

The sum of n independent Uniform(0,1) random variables follows the Irwin-Hall distribution. For n=2, the "triangular" distribution. For n=12, it closely approximates the standard normal — this was historically used to generate normal random numbers before more efficient algorithms existed.

Frequently Asked Questions

• When should I use the uniform distribution?

  Use it when the model really implies equal likelihood across the interval or set: fair dice, random number generation, simple timing assumptions, and other cases where you do not want to weight one value more than another.

• What's the difference between continuous and discrete uniform?

  Continuous uniform applies to real numbers in interval [a,b] — any value is possible. Discrete uniform applies to integers from a to b, each with equal probability. A die roll is discrete; a random time is continuous.

• Why is the variance (b−a)²/12 and not (b−a)²/4?

  The denominator 12 comes from integration. The variance of U(0,1) is the integral of (x − 0.5)² from 0 to 1, which equals 1/12. Scaling by (b−a) gives (b−a)²/12. The factor of 12 is fundamental, not arbitrary.

• What are order statistics?

  If you take n samples from a distribution and sort them, the k-th smallest is the k-th order statistic X₍ₖ₎. For uniform distributions, E[X₍ₖ₎] divides the interval into n+1 equal parts, placing the k-th value at position k/(n+1).

• How does the sampling distribution work?

  The mean of n uniform samples has mean (a+b)/2 (same as individual) but standard error σ/√n. By the CLT, the sample mean is approximately normal for large n, even though individual values are uniform.

• Can the bounds be negative?

  Yes! U(−5, 5) is perfectly valid. The mean would be 0, variance 100/12 ≈ 8.33. Negative bounds are common when modeling measurement errors centered around zero.