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For a random variable following the continuous uniform distribution, the expected value is = +, and the variance is = (). For the special case a = − b , {\displaystyle a=-b,} the probability density function of the continuous uniform distribution is:
Less simply, a random permutation is a permutation generated uniformly randomly from the permutations of a given set and a uniform spanning tree of a graph is a spanning tree selected with uniform probabilities from the full set of spanning trees of the graph. The discrete uniform distribution itself is non-parametric.
The uniform distribution or rectangular distribution on [a,b], where all points in a finite interval are equally likely, is a special case of the four-parameter Beta distribution. The Irwin–Hall distribution is the distribution of the sum of n independent random variables, each of which having the uniform distribution on [0,1].
A chart showing a uniform distribution. In probability theory and statistics, a collection of random variables is independent and identically distributed (i.i.d., iid, or IID) if each random variable has the same probability distribution as the others and all are mutually independent. [1]
Indeed, even when the random variable does not have a density, the characteristic function may be seen as the Fourier transform of the measure corresponding to the random variable. Another related concept is the representation of probability distributions as elements of a reproducing kernel Hilbert space via the kernel embedding of distributions .
The distribution of the product of a random variable having a uniform distribution on (0,1) with a random variable having a gamma distribution with shape parameter equal to 2, is an exponential distribution. [18]
In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to random variables having a standard uniform distribution. [1]
An absolutely continuous random variable is a random variable whose probability distribution is absolutely continuous. There are many examples of absolutely continuous probability distributions: normal , uniform , chi-squared , and others .