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The Bates distribution is sometimes confused [2] with the Irwin–Hall distribution, which is the distribution of the sum (not the mean) of n independent random variables uniformly distributed from 0 to 1. If X has a Bates distribution on the unit interval, then n X has an Irwin-Hall distribution; when n = 1 they are both uniformly distributed.
If X has a standard uniform distribution, then Y = X n has a beta distribution with parameters (1/n,1). As such, The Irwin–Hall distribution is the sum of n i.i.d. U(0,1) distributions. The Bates distribution is the average of n i.i.d. U(0,1) distributions. The standard uniform distribution is a special case of the beta distribution, with ...
The Bates distribution is the distribution of the mean of n independent random variables, each of which having the uniform distribution on [0,1]. The logit-normal distribution on (0,1). The Dirac delta function , although not strictly a probability distribution, is a limiting form of many continuous probability functions.
By the Central Limit Theorem, as n increases, the Irwin–Hall distribution more and more strongly approximates a Normal distribution with mean = / and variance = /.To approximate the standard Normal distribution () = (=, =), the Irwin–Hall distribution can be centered by shifting it by its mean of n/2, and scaling the result by the square root of its variance:
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.
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Here the problem of defining or manipulating a joint probability distribution for a set of random variables is simplified or reduced in apparent complexity by applying the probability integral transform to each of the components and then working with a joint distribution for which the marginal variables have uniform distributions.
For example, the drawings of a random variable uniform over a segment will be equidistributed in the segment, but there will be large gaps compared to a sequence which first enumerates multiples of ε in the segment, for some small ε, in an appropriately chosen way, and then continues to do this for smaller and smaller values of ε.