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Suppose we select a point X at random from an interval [0, 1], where the probability that a point is in any subinterval of [0, 1] only depends on the length of the subinterval. X has a standard uniform distribution, with a constant probability density of 1 over the interval [0, 1] and zero everywhere else.

Probability Density Function

f ( x ) = 1

Support

0 ≤ x ≤ 1

Mean

Variance

Example
A number chosen at random and with equal probability from anywhere in the interval [0, 1] has a standard uniform distribution.
You arrive at a bus stop for a bus that runs every hour, but don't know when the previous bus left. The time spent waiting for the next bus has a standard uniform distribution.
A spinner is spun randomly until it comes to rest. The position of the spinner as a fraction of a full circle has a standard uniform distribution.

X ~ Standard Uniform

Chart of the standard uniform distribution Chart area for displaying the standard uniform pdf, cdf, visualization, and simulation

E(X) = , Var(X) =

The standard uniform distribution is central to random number generation. If U has a standard uniform distribution and X has cdf FX(x), then FX-1(U) will have the distribution FX. In this way, we can generate a random sample from any distribution for which FX-1 is known, given a standard uniform random sample.

The illustration above shows a point X chosen randomly and with uniform probability from the interval [0, 1]. If the probability that X lies in any subinterval I equals the length of I, then X is a standard uniform random variable.

The simulation above shows a point X chosen randomly and with uniform probability from the interval [0, 1]. The histogram accumulates the results of each simulation.

Y = max Xᵢ ~ Beta(n, 1) min Xᵢ ~ Beta(1, n) X(j) ~ Beta(j, n - j + 1) Xⁿ ~ Beta(1/n, 1) -βlog(1 - X) ~ Exponential(β) log(X₁/X₂) ~ Laplace(0, 1) tan(π(X - ½)) ~ Standard Cauchy log(X/(1 - X)) ~ Standard Logistic 1/X ~ Standard Pareto a + (b - a)X ~ Uniform(a, b)

Chart of the related distribution Chart area for displaying the related pdf, cdf, visualization, and simulation

E(Y) = , Var(Y) =

Note that as n increases, the probability mass moves to the right. This occurs because higher values become more likely when finding the maximum of a larger sample.

Note that as n increases, the probability mass moves to the left. This occurs because lower values become more likely when finding the minimum of a larger sample.

The graph above shows the distribution of the median X((n + 1)/2) for odd n. Note that as n increases the distribution becomes more sharply peaked around 0.5.

Note that the median is located at 1/2n. So, as n increases, the median decreases and the probability mass moves to the left.

Note that -βlog(1 - X) is the inverse cdf FX-1(X) of the exponential(β) distribution. When a standard uniform random variable is transformed by the inverse cdf of another random variable X, the transformed random variable has the distribution of X.

Since log(X₁/X₂) = log X₁ - log X₂, and since -log X is an exponential(1) random variable when X is standard uniform, then log(X₁/X₂) is the difference of two exponential(1) random variables. This can be proven to be a Laplace(1) random variable.

Note that tan(π (X - 1/2)) is the inverse cdf FX-1(X) of the standard Cauchy distribution. When a standard uniform random variable is transformed by the inverse cdf of another random variable X, the transformed random variable has the distribution of X.

Y can be visualized as the height of a right-angled triangle with base length 1, where the angle θ opposite the height varies uniformly in the range [-π/2, π/2].

Note that log[X/(1 - X)] is the inverse cdf FX-1(X) of the standard logistic distribution. When a standard uniform random variable is transformed by the inverse cdf of another random variable X, the transformed random variable has the distribution of X.

Note that: (i) 1 - X is also a standard uniform random variable, so 1/(1 - X) has the same distribution as 1/X; (ii) 1/(1 - X) is the inverse cdf FX-1(X) of the standard Pareto distribution. When a standard uniform random variable is transformed by the inverse cdf of another random variable X, the transformed random variable has the distribution of X. Hence, Y = 1/X is a standard Pareto random variable

This follows since the uniform distribution is a location-scale family of distributions.