Random variables, transformations of

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

Example 1. Let $ X $ be a random variable having a continuous and strictly increasing distribution function $ F $. Then the random variable $ Y = F ( X) $ has a uniform distribution on the interval $ [ 0 , 1 ] $, and the random variable $ Z = \Phi ^ {-1} ( F ( X) ) $( where $ \Phi $ is the standard normal distribution function) has a normal distribution with parameters 0 and 1. Conversely, the formula $ X = F ^ { - 1 } ( \Phi ( Z) ) $ enables one to obtain a random variable $ X $ that has the given distribution function $ F $ from a random variable $ Z $ 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 $ Z _ {n} $ be asymptotically normal with parameters $ ( 0 , 1 ) $. One then poses the problem of constructing simple (and simply invertible) functions $ f _ {n} $ such that the random variables $ V _ {n} = Z _ {n} + f _ {n} ( Z _ {n} ) $ are "more normal" than $ Z _ {n} $.

Example 2. Let $ X _ {1} \dots X _ {n} \dots $ be independent random variables, each having a uniform distribution on $ [ - 1 , 1 ] $, and put

$$ Z _ {n} = \frac{X _ {1} + \dots + X _ {n} }{\sqrt {n / 3 } } . $$

By the central limit theorem,

$$ {\mathsf P} \{ Z _ {n} < x \} - \Phi ( x) = O \left ( \frac{1}{n} \right ) . $$

If one sets

$$ V _ {n} = Z _ {n} - \frac{1}{20n} ( 3 Z _ {n} - Z _ {n} ^ {3} ) , $$

then

$$ {\mathsf P} \{ V _ {n} < x \} - \Phi ( x) = O \left ( \frac{1}{n ^ {2} } \right ) . $$

Example 3. The random variables $ \chi _ {n} ^ {2} $, $ \sqrt {2 \chi _ {n} ^ {2} } $ and $ ( \chi _ {n} ^ {2} / n ) ^ {1/3} $ are asymptotically normal as $ n \rightarrow \infty $( see Chi-squared distribution). The uniform deviation of the corresponding distribution functions from their normal approximations becomes less than $ 0 . 0 1 $ for $ \chi _ {n} ^ {2} $ when $ n \geq 354 $, and for $ \sqrt {2 \chi _ {n} ^ {2} } $( the Fisher transformation) — when $ n \geq 23 $; for $ ( \chi _ {n} ^ {2} / n ) ^ {1/3} $( the Wilson–Hilferty transformation) when $ n \geq 3 $ this deviation does not exceed $ 0.0007 $.

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" ).

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Related to the transformations above are the Edgeworth expansions (see, e.g., [a1]; cf. also Edgeworth series).

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