Uniform distribution (continuous)

The continuous uniform distribution, also called the rectangular distribution or uniform distribution is a continuous probability distribution. She has a constant probability density on an interval. This is equivalent to saying that all subintervals of equal length have the same probability.

Definition

A continuous random variable is uniformly distributed as designated on the interval when the density function and distribution function are given as

As a shorthand notation for the continuous uniform distribution is frequently or used. In some formulas can also be seen as a term or for distribution.

Properties

Probabilities

The probability that a uniformly distributed random variable lies on a sub-interval corresponds to the ratio of the interval lengths:

Mean and median

The expected value and the median of the continuous uniform distribution corresponding to the midpoint of the interval:

Variance

Is the variance of the constant uniform distribution

Standard deviation and other measures of variation

From the variance we obtain the standard deviation

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

Coefficient of variation

For the coefficient of variation results in:

Skew

The skewness can be represented as

Curvature and excess

The curvature and the excess can also be represented as a closed

Moments

Sum of equally distributed random variables

The sum of two independent uniformly distributed random variables and steadily the same carrier - width is a triangular distribution, otherwise there is a trapezoidal distribution. More precisely:

Two random variables are equally distributed independently and steadily, the one on the interval, and the other on the interval. Be and. Then your sum has the following distribution:

The sum of independent identically distributed random variables approaches the normal distribution ( Central Limit Theorem ).

One method sometimes used ( Twelver rule) for the approximate generation ( standard ) normally distributed random numbers works like this: you summed 12 (independent) on the interval [ 0,1] uniformly distributed random numbers and subtracts 6 ( which provides the right moments, since the variance of a U ( 0,1) random variable 1/12, and it has the expected value of 1/2).

Characteristic function

The characteristic feature is in the form

The imaginary number represents.

Moment generating function

The moment generating function of the continuous uniform distribution is

And specifically for and

Relationship to other distributions

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

For example for the interval [0,1]

Will be accepted and often, so considered. Then, the density distribution function and the function on the interval [ 0,1]. The expected value of E (X) = 0.5 and the variance Var ( X) = 1/12, thus the standard deviation. See also the section above sum of equally distributed random variables.

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