Uniform Distribution Calculator

Compute probabilities, PDF, CDF, quantiles, and measures for the continuous uniform distribution

About the Uniform Distribution Calculator

The Uniform Distribution Calculator is a statistical tool designed to evaluate continuous random variables where every outcome within a specific interval is equally likely. This distribution, often represented as U(a, b), is fundamental in fields ranging from computer simulation and physics to logistics and finance. It is characterized by its constant probability density, which creates a rectangular shape when graphed, indicating that the likelihood of any sub-interval depends solely on its length rather than its location.

Engineers and data scientists use this tool to model phenomena with known boundaries but no central tendency, such as the wait time for a shuttle that arrives at perfectly regular intervals or the rounding error in numerical calculations. By inputting the minimum and maximum parameters, users can instantly determine the probability of a value falling within a specific range, calculate the cumulative distribution function (CDF), and find essential descriptive statistics like the mean, variance, and standard deviation. This calculator simplifies the process of analyzing "balanced" risks where no outcome is more favored than another.

Formula

P(x1 < X < x2) = (x2 - x1) / (b - a) for a ≤ x1 < x2 ≤ b

The probability of a value falling between two points (x1 and x2) is the length of that interval divided by the total range of the distribution. The total range is defined by the lower bound (a) and the upper bound (b).

The Probability Density Function (PDF) is f(x) = 1 / (b - a) for all values within the range, ensuring the area under the rectangle equals 1. The Mean (expected value) is calculated as (a + b) / 2, representing the exact midpoint of the distribution.

Worked examples

Example 1: A bus arrives every 10 minutes (a=0, b=10). What is the probability it arrives between the 3rd and 7th minute?

1. Identify bounds: a = 0, b = 10.
2. Identify target interval: x1 = 3, x2 = 7.
3. Calculate interval length: 7 - 3 = 4.
4. Calculate total range: 10 - 0 = 10.
5. Divide interval by range: 4 / 10 = 0.4.

Result: 0.40 or 40%. There is a 40% chance the bus arrives in that specific 4-minute window.

Example 2: A factory machine's efficiency is uniformly distributed between 50% and 100%. Find the mean and variance.

1. Set a = 50, b = 100.
2. Mean = (50 + 100) / 2 = 75.
3. Variance = (100 - 50)^2 / 12.
4. Variance = 50^2 / 12 = 2500 / 12 = 208.33.

Result: Mean = 75, Variance = 208.33. The average success rate is 75% with a broad spread.

Common use cases

Estimating the probability of a customer arriving during a specific 10-minute window of an hour-long lunch shift.
Calculating the expected rounding error when measurements are rounded to the nearest millimeter.
Modeling the time remaining before a scheduled light rail arrival when the passenger arrives at a random moment.
Determining the likelihood of a chemical reaction occurring within a specific temperature range in a stable environment.

Pitfalls and limitations

Entering a value for x that is outside the bounds of a and b will result in a probability of 0 or 1.
Confusing the continuous uniform distribution with the discrete version, which applies to countable outcomes like a die roll.
Applying uniform distribution to data that actually follows a bell curve or has outliers.
Forgetting that the probability of the random variable equaling an exact specific number is always zero in continuous distributions.

Frequently asked questions

why is uniform distribution called rectangular distribution

Uniform distributions are often called rectangular distributions because the height of the probability density function is constant between the minimum and maximum boundaries, forming a rectangle on a graph.

can uniform distribution probability be zero at a point

Yes, if the random variable represents a duration, the probability of it being exactly a single value (like exactly 5.000... minutes) is zero. You must calculate the probability over an interval or use the cumulative distribution function.

what is the standard uniform distribution formula

The standard uniform distribution is a special case where the lower bound (a) is 0 and the upper bound (b) is 1, often used in computer science for generating random numbers.

uniform distribution vs normal distribution difference

Unlike the normal distribution, the uniform distribution has no central peak; every interval of a fixed length within the bounds is equally likely to contain an observation.