Real random variable

Random function

A random function is a function of some Real random variable , or rather a function that composes with the random variable to create a function on the sample space . prob

Distribution

The probability density function of a random function is given by prob

where is the Dirac delta.¹

Proof

Let and be a random function. Then the Characteristic function (probability) of is

Applying the inverse Fourier transform:

Now using the Fourier representation of the Dirac delta

This expands to multivariate scenarios as expected.

In the discrete case the probability mass function is

See also

• Distribution of a differentiable injective random function

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Footnotes

1. 2006, Statistische Mechanik, p. 5 ↩