Uniform Distribution Calculator
Calculate mean, variance, PDF, and CDF of a continuous uniform distribution on [a, b].
Enter any x to find the probability P(X ≤ x) instantly.
The continuous uniform distribution U(a, b) describes a random variable that is equally likely to take any value between a and b, and impossible outside that range.
PDF: f(x) = 1/(b-a) for a ≤ x ≤ b, and 0 everywhere else
CDF: F(x) = P(X ≤ x) = (x-a)/(b-a) for a ≤ x ≤ b
Mean: μ = (a+b)/2
Variance: σ² = (b-a)²/12
Standard deviation: σ = (b-a)/√12 = (b-a)/(2√3)
Median: (a+b)/2 (same as the mean — the distribution is symmetric)
The PDF is a flat horizontal line at height 1/(b-a). The CDF is a straight line from 0 at x=a to 1 at x=b.
When to use it
The uniform distribution applies when all outcomes in a range are equally likely and you have no reason to prefer any particular value. A random number generator that produces values from 0 to 1 follows U(0,1). The classic “roll a fair die” situation is a discrete uniform distribution; rolling a spinner on a continuous scale is continuous uniform.
It is also the prior in Bayesian inference when you have no prior information about a parameter — a “flat prior” says you find all values equally plausible before seeing data.
Key interval probability
P(x₁ < X < x₂) = (x₂ - x₁)/(b - a) for any x₁, x₂ within [a, b]. The fraction of the total interval determines the probability. If the interval is (2, 5) and the range is (0, 10), there is a 30% chance X falls in that interval.
The standard uniform U(0,1) is a special case used constantly in simulation and random number generation.