Uniform Distribution
Calculate uniform distribution PDF, CDF, mean, variance, and standard deviation for U(a,b). Free online uniform distribution probability calculator with charts and step-by-step results.
About This Calculator
The Uniform Distribution Calculator helps you compute probabilities, density values, and key statistics for any continuous uniform distribution U(a,b). Whether you are a student learning probability theory, a statistician analyzing equally-likely outcomes, or a data scientist generating random variates, this tool provides instant results for PDF, CDF, mean, median, variance, and standard deviation.
Methodology
The continuous uniform distribution is defined over an interval [a,b] where every sub-interval of equal length has the same probability. The probability density function is constant: f(x) = 1/(b-a) for a ≤ x ≤ b. The cumulative distribution function increases linearly: F(x) = (x-a)/(b-a). Because the distribution is perfectly symmetric, its mean equals its median at (a+b)/2, and its skewness is exactly zero. The variance is (b-a)²/12 and the standard deviation is (b-a)/√12.
Regional Notes
India: Uniform distribution is fundamental in the CBSE and university statistics curriculum for probability and random variable analysis. It appears in fields ranging from quality control in manufacturing to Monte Carlo simulations in finance.
US: Widely used in AP Statistics, college-level probability courses, and data science applications. The standard uniform distribution U(0,1) is the basis for random number generation in R, Python, and Excel.
UK: A core topic in A-Level Mathematics and Further Statistics, the uniform distribution is applied in operational research, risk modeling, and actuarial science. The rectangular distribution is another common name in British textbooks.
Frequently Asked Questions
What is a uniform distribution?
A uniform distribution is a probability distribution where all outcomes in an interval [a,b] are equally likely. Its probability density function is constant (f(x) = 1/(b-a)) and its graph forms a rectangle, which is why it is also called the rectangular distribution.
How do I calculate the mean of a uniform distribution?
The mean of a continuous uniform distribution U(a,b) is the midpoint of the interval: μ = (a + b)/2. For example, U(0,10) has a mean of 5, and U(0,1) has a mean of 0.5.
What is the variance of a uniform distribution?
The variance of a uniform distribution U(a,b) is σ² = (b-a)²/12 and the standard deviation is σ = (b-a)/√12. For U(0,10), the standard deviation is approximately 2.8868.
Is uniform distribution the same as normal distribution?
No, they are different distributions. The uniform distribution has constant probability over an interval with a rectangular-shaped graph. The normal distribution is bell-shaped with outcomes near the mean more likely. The uniform distribution has skewness of zero and no tails, while the normal distribution approaches zero asymptotically.
What is the PDF formula for uniform distribution?
The probability density function (PDF) of a continuous uniform distribution U(a,b) is f(x) = 1/(b-a) for a ≤ x ≤ b and f(x) = 0 elsewhere. The CDF is F(x) = (x-a)/(b-a) for a ≤ x ≤ b, with F(x) = 0 for x less than a and F(x) = 1 for x greater than b.
What is the difference between discrete and continuous uniform distribution?
A discrete uniform distribution has finitely many equally likely outcomes (like rolling a fair die with P(x)=1/6 for each face). A continuous uniform distribution has infinitely many outcomes over an interval [a,b] where the probability of any exact value is zero but probabilities over sub-intervals are proportional to length.
How do I calculate P(c ≤ X ≤ d) for a uniform distribution?
For a uniform distribution U(a,b), the probability that X falls between c and d (where a ≤ c ≤ d ≤ b) is P(c ≤ X ≤ d) = (d-c)/(b-a). This equals the area of the rectangle with width (d-c) and height 1/(b-a).
What is the quantile function of uniform distribution?
The quantile function (inverse CDF) of the uniform distribution U(a,b) is Q(p) = (b-a)·p + a for 0 ≤ p ≤ 1. For the standard uniform distribution U(0,1), the quantile function is simply Q(p) = p.