Random function

A random function $𝑇$ is a function of some Real random variable $𝑋$ , or rather a function that composes with the random variable $𝑋$ to create a function on the sample space $𝑇 \circ 𝑋 : 𝜉 \to 𝑆$ . #m/def/prob

Distribution

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

𝐹 𝑇 (𝑡) = 𝔼 [𝛿 (𝑇 (𝑋) - 𝑡)]

where $𝛿$ is the Dirac delta.¹

Proof

Let $𝑋 \sim 𝑤$ and $𝐹 (𝑋)$ be a random function. Then the Characteristic function (probability) of $𝐹$ is

𝜒 𝐹 (𝑘) = \infty \sum 𝑛 = 0 (- 𝑖 𝑘) 𝑛 𝑛! ⟨ 𝐹 (𝑋) 𝑛 ⟩

Applying the inverse Fourier transform:

𝑤 𝐹 (𝑓) = 1 2 𝜋 \int \infty - \infty (\infty \sum 𝑛 = 0 (- 𝑖 𝑘) 𝑛 𝑛! ⟨ 𝐹 𝑛 ⟩) 𝑒 𝑖 𝑘 𝑓 𝑑 𝑘 = 1 2 𝜋 \int \infty - \infty (\infty \sum 𝑛 = 0 (- 𝑖 𝑘) 𝑛 𝑛! \int \infty - \infty 𝐹 (𝑥) 𝑛 𝑤 (𝑥) 𝑑 𝑥) 𝑒 𝑖 𝑘 𝑓 𝑑 𝑘 = 1 2 𝜋 \int \infty - \infty (\int \infty - \infty (\infty \sum 𝑛 = 0 (- 𝑖 𝑘) 𝑛 𝑛! 𝐹 (𝑥) 𝑛) 𝑤 (𝑥) 𝑑 𝑥) 𝑒 𝑖 𝑘 𝑓 𝑑 𝑘 = 1 2 𝜋 \int \infty - \infty 𝑒 𝑖 𝑘 𝑓 \int \infty - \infty 𝑒 - 𝑖 𝑘 𝐹 (𝑥) 𝑤 (𝑥) 𝑑 𝑥 𝑑 𝑘 = 1 2 𝜋 \int \infty - \infty \int \infty - \infty 𝑒 𝑖 𝑘 (𝑓 - 𝐹 (𝑥)) 𝑑 𝑘 𝑤 (𝑥) 𝑑 𝑥

Now using the Fourier representation of the Dirac delta

𝑤 𝐹 (𝑓) = \int \infty - \infty 𝛿 (𝑓 - 𝐹 (𝑥)) 𝑤 (𝑥) 𝑑 𝑥 = ⟨ 𝛿 (𝑓 - 𝐹 (𝑥)) ⟩

This expands to multivariate scenarios as expected.

In the discrete case the probability mass function is

𝑝 𝑇 (𝑡) = \sum 𝑥 : 𝑇 (𝑥) = 𝑦 𝑝 𝑋 (𝑦)

Random function

Distribution

Footnotes