# Random variables, transformations of

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The determination of functions of given arbitrary random variables for which the probability distributions possess given properties.

Example 1. Let be a random variable having a continuous and strictly increasing distribution function . Then the random variable has a uniform distribution on the interval , and the random variable (where is the standard normal distribution function) has a normal distribution with parameters 0 and 1. Conversely, the formula enables one to obtain a random variable that has the given distribution function from a random variable with a standard normal distribution.

Transformations of random variables are often used in connection with limit theorems of probability theory. For example, let a sequence of random variables be asymptotically normal with parameters . One then poses the problem of constructing simple (and simply invertible) functions such that the random variables are "more normal" than .

Example 2. Let be independent random variables, each having a uniform distribution on , and put

By the central limit theorem,

If one sets

then

Example 3. The random variables , and are asymptotically normal as (see "Chi-squared" distribution). The uniform deviation of the corresponding distribution functions from their normal approximations becomes less than for when , and for (the Fisher transformation) — when ; for (the Wilson–Hilferty transformation) when this deviation does not exceed .

Transformations of random variables have long been applied in problems of mathematical statistics as the basis for constructing simple asymptotic formulas of high precision. Transformations of random variables are also used in the theory of stochastic processes (for example, the method of the "single probability space" ).

#### References

 [1] L.N. Bol'shev, "On transformations of random variables" Theory Probab. Appl. , 4 (1959) pp. 129–141 Teor. Veryatnost. Primenen. , 4 : 2 (1959) pp. 136–149 [2] L.N. Bol'shev, "Asymptotically Pearson transformations" Theory Probab. Appl. , 8 : 2 (1963) pp. 121–146 Teor. Veroyatnost. Primenen. , 8 : 2 (1963) pp. 129–155 [3] L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)