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The continuous uniform distribution with parameters = and =, i.e. (,), is called the standard uniform distribution. One interesting property of the standard uniform distribution is that if u 1 {\displaystyle u_{1}} has a standard uniform distribution, then so does 1 − u 1 . {\displaystyle 1-u_{1}.}
The general formula for the variance of the outcome, X, of an n-sided die is ... Uniform distribution (continuous) ...
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].
The formula in the definition of characteristic function allows us to compute φ when we know the distribution function F (or density f). If, on the other hand, we know the characteristic function φ and want to find the corresponding distribution function, then one of the following inversion theorems can be used. Theorem.
This substantially unifies the treatment of discrete and continuous probability distributions. The above expression allows for determining statistical characteristics of such a discrete variable (such as the mean, variance, and kurtosis), starting from the formulas given for a continuous distribution of the probability.
Any probability distribution can be decomposed as the mixture of a discrete, an absolutely continuous and a singular continuous distribution, [14] and thus any cumulative distribution function admits a decomposition as the convex sum of the three according cumulative distribution functions.
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
It is also the continuous distribution with the maximum entropy for a specified mean and variance. [18] [19] Geary has shown, assuming that the mean and variance are finite, that the normal distribution is the only distribution where the mean and variance calculated from a set of independent draws are independent of each other. [20] [21]