# Distribution, Uniform

# Distribution, Uniform

Uniform distribution is the probability distribution in which the probability is uniform for all intervals of the same length. The definition of the continuous uniform distribution function contains two parameters, *a* and *b*, which are the minimum and maximum values, respectively, that can occur in the set of numbers characterized by the distribution. The probability of any number in the interval (*a, b* ) is 1/(*b* – *a* ), and 0 outside that interval, as illustrated in Figure 1. Whether *a* and *b* are included in the interval does not matter to the integral, and in practice are sometimes included and sometimes not. The distribution is also known as rectangular distribution, because of its rectangular shape. As in all probability density functions, the area under the curve is 1. The standard uniform distribution is the special case where *a* is 0 and *b* is 1, so that the distribution takes the form of a square of height 1. In a probability distribution, the area of the curve under any interval represents the probability that a number described by the distribution occurs in that interval. For the standard uniform distribution, the probability of seeing a number in the interval (0, 0.5) would be the same as the probability of seeing a number in the interval (0.5, 1), and both would be 50 percent. In the discrete form of the uniform distribution, the probability of occurrence of all values in a finite set is equal. For example, if the set of possible values were 0 and 1, then there would be a 50 percent chance for 0 and a 50 percent chance for 1; or a 50 percent chance for getting heads and a 50 percent chance of getting tails on a fair coin.

Uniformly distributed phenomena are rare in the social world, because uniform probability implies randomness, and the social world is characterized by pattern, or the lack of randomness. However, social scientists still find use for the uniform distribution; in fact, it is the second-most used distribution after the normal distribution. The most popular use of the uniform distribution is to find the random variates of other probability distributions, such as the normal distribution. A random variate of a distribution is a number chosen randomly out of the set of numbers with likelihoods characterized by the distribution. A uniform normal variate input to an inverse cumulative probability function of any distribution will generate a random variate of that distribution. Random variates are important in Monte Carlo simulation studies, where the distribution of an outcome is estimated by taking many samples in many simulation runs. Ironically,

truly random variates of any distribution cannot be computed, because truly random numbers cannot be expressed in an equation by definition. Therefore, pseudorandom numbers are accepted, usually created by modulo arithmetic, which have many of the statistical properties of random numbers but repeat themselves at some point. True random numbers are normal in the sense that, for each *k*, all subsequences of the binary (or decimal, or other) expansion of length *k* have equal probability, but pseudorandom numbers fail on this requirement for some *k*.

Social scientists also use the uniform distribution to represent lack of knowledge. For example, in a simulation where a distribution is not known, uniform random variates are often used. Naturally, the uniform random variate will incorrectly represent the underlying distribution. However, the uniform distribution also represents independence. The random variates of any distribution incorrectly represent dependencies on the random variates of other modeled distributions. A random variate is only a good model when the measure represents phenomena that are independent of the other phenomena being represented. True independence is as rare as true randomness in the social world. A simple illustration of the problem with using the uniform distribution to represent the lack of knowledge is Bertrand’s paradox. In Bertrand s paradox, a cube is hidden in a box with a side that has an unknown length, say between 3 and 5 centimeters. Using modulo arithmetic can then generate random numbers between 3 and 5 centimeters, and take their average in multiple runs of 4 centimeters to estimate the side length. Random numbers could also be generated for all the possible surface areas, between 54 and 150 square centimeters, and all the possible volumes, between 27 and 125 cubic centimeters. However, if those possible measures are averaged as well, then an impossible cube emerges with a length of 4 centimeters, a surface area of 102 square centimeters, and a volume of 76 cubic centimeters.

**SEE ALSO** *Distribution, Normal; Distribution, Poisson; Frequency Distributions; Monte Carlo Experiments; Probability; Variables, Random*

## BIBLIOGRAPHY

Chaitin, Gregory. 1975. Randomness and Mathematical Proof. *Scientific American* 232 (5): 47– *52*.

Clarke, Michael. 2002. *Paradoxes from A to Z*. London: Routledge.

Jaynes, Edwin Thompson. 1973. *The Well-Posed Problem*. Foundations of Physics, vol. 3, pp. 477–493.

Jaynes, Edwin Thompson. 2003. *Probability Theory: The Logic of Science*. Cambridge University Press, 2003.

*Deborah Vakas Duong*

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