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 . #m/def/prob

Distribution

The probability density function of a random function is given by #m/thm/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

#state/tidy | #lang/en | #SemBr

Footnotes

  1. 2006, Statistische Mechanik, p. 5 ↩