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
Support
Mean
Variance
Example |
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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
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 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.