# Constant equal distribution

The **constant uniform distribution** , also called **rectangular ****distribution** , **continuous uniform distribution** , or **uniform ****distribution** , is a constant probability distribution . It has a constant probability density over an interval . This is synonymous with the fact that all sub-intervals of the same length have the same probability.

The possibility of simulating the constant uniform distribution on the interval from 0 to 1 forms the basis for generating numerous randomly distributed random numbers using the inversion method or the rejection method .

## definition

A continuous random variable is called **uniformly distributed** over the interval if the density function and distribution function are given as

Often or is used as an abbreviation for the constant uniform distribution . In some formulas one also sees or as a designation for the distribution. The constant uniform distribution is completely described by its first two central moments, i.e. H. all higher moments can be calculated from expected value and variance.

## properties

### Probabilities

The probability that an evenly distributed random variable lies in a sub-interval is equal to the ratio of the interval lengths:

- .

### Expected value and median

The expected value and the median of the constant uniform distribution are equal to the middle of the interval :

- .

### Variance

The variance of the constant uniform distribution is

### Standard deviation and other measures of variance

The standard deviation is obtained from the variance

- .

The mean absolute deviation is , and the interquartile range is exactly twice as large. The uniform distribution is the only symmetrical distribution with monotonic density with this property.

### Coefficient of variation

The following results for the coefficient of variation :

- .

### symmetry

The constant uniform distribution is symmetrical around .

### Crookedness

The skew can be represented as

- .

### Bulge and excess

The bulge and the excess can also be represented as closed

- or.
- .

### Moments

-th moment | |

-th central moment |

### Sum of uniformly distributed random variables

The sum of two independent and continuously uniformly distributed random variables with the same carrier width is triangularly distributed , otherwise a trapezoidal distribution results. More accurate:

Let two random variables be independent and continuously evenly distributed, one on the interval , the other on the interval . Be and . Then their sum has the following trapezoidal distribution :

The sum of independent, uniformly distributed random variables on the interval [0; 1] is an Irwin-Hall distribution ; it approximates the normal distribution ( central limit theorem ).

An occasionally used method ( rule of twelve ) for the approximate generation of (standard) normally distributed random numbers works like this: one adds up 12 (independent) random numbers uniformly distributed over the interval [0,1] and subtracts 6 (this provides the correct moments, since the variance of a U (0,1) -distributed random variable is 1/12 and it has the expectation value 1/2).

### Characteristic function

The characteristic function has the form

- ,

where represents the imaginary unit .

### Moment generating function

The moment-generating function of the constant uniform distribution is

and especially for and

## Relationship to other distributions

### Relationship to the triangular distribution

The sum of two independent and continuously equally distributed random variables has a triangular distribution .

### Relationship to beta distribution

If there are independent random variables that are constantly uniformly distributed, then the order statistics have a beta distribution . More precisely applies

for .

### Simulation of distributions from the constant uniform distribution

With the inversion method, uniformly distributed random numbers can be converted into other distributions. If is a uniformly distributed random variable, then, for example, the exponential distribution with the parameter is sufficient .

### Generalization to higher dimensions

The continuous uniform distribution can be from the interval in any measurable subsets of with Lebesgue measure generalize. Then you bet

for measurable .

### Discreet case

The uniform distribution is also defined on finite sets, then it is called discrete uniform distribution .

## Example for the interval [0, 1]

Often it is and assumed, that is, considered. Then the density function on the interval is constant equal to 1 and the distribution function there applies . The expected value is accordingly , the variance and the standard deviation , whereby the latter two values also apply to *any* intervals of length 1. See also the section above, Sum of uniformly distributed random variables .

Is a -distributed random variable, then is

-distributed.

## See also

## literature

- Christian Hesse : Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , pp. 155-156 , doi : 10.1007 / 978-3-663-01244-3 .

## Web links

- Interactive animation - University of Konstanz