Such a distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.
It is the maximum entropy probability distribution for a random variable
[3] The probability density function of the continuous uniform distribution is The values of
so the probability density function of the continuous uniform distribution is graphically portrayed as a rectangle where
As the base length increases, the height (the density at any particular value within the distribution boundaries) decreases.
In a graphical representation of the continuous uniform distribution function
the area under the curve within the specified bounds, displaying the probability, is a rectangle.
The example above is a conditional probability case for the continuous uniform distribution: given that
Conditional probability changes the sample space, so a new interval length
[5] The graphical representation would still follow Example 1, where the area under the curve within the specified bounds displays the probability; the base of the rectangle would be
[5] The moment-generating function of the continuous uniform distribution is:[6] from which we may calculate the raw moments
For a random variable following the continuous uniform distribution, the expected value is
be a Borel set of positive, finite Lebesgue measure
It is possible to obtain a uniform distribution on the standard n-vertex simplex in the following way.
This problem is commonly known as the German tank problem, due to application of maximum estimation to estimates of German tank production during World War II.
with unknown a, the maximum likelihood estimator for a is: the sample minimum.
Note that the interval length depends upon the random variable
[2] Therefore, there are various applications that this distribution can be used for as shown below: hypothesis testing situations, random sampling cases, finance, etc.
Furthermore, generally, experiments of physical origin follow a uniform distribution (e.g. emission of radioactive particles).
[1] However, it is important to note that in any application, there is the unchanging assumption that the probability of falling in an interval of fixed length is constant.
[2] In the field of economics, usually demand and replenishment may not follow the expected normal distribution.
As a result, other distribution models are used to better predict probabilities and trends such as Bernoulli process.
[11] But according to Wanke (2008), in the particular case of investigating lead-time for inventory management at the beginning of the life cycle when a completely new product is being analyzed, the uniform distribution proves to be more useful.
[11] The uniform distribution would be ideal in this situation since the random variable of lead-time (related to demand) is unknown for the new product but the results are likely to range between a plausible range of two values.
It was also noted that the uniform distribution was also used due to the simplicity of the calculations.
Since simulations using this method require inverting the CDF of the target variable, alternative methods have been devised for the cases where the CDF is not known in closed form.
The normal distribution is an important example where the inverse transform method is not efficient.
However, there is an exact method, the Box–Muller transformation, which uses the inverse transform to convert two independent uniform random variables into two independent normally distributed random variables.
On the other hand, the uniformly distributed numbers are often used as the basis for non-uniform random variate generation.
Equiprobability was mentioned in Gerolamo Cardano's Liber de Ludo Aleae, a manual written in 16th century and detailed on advanced probability calculus in relation to dice.