Lagmental Vicfred

A Distribution Is the Pushforward of Probability by Vicfred

tags: mathematics, conditional probability, random variables, distributions

The law of a random variable assigns each measurable set the probability of its inverse image. This is a compact note, but the quantifiers and hypotheses stay on the page.

Definitions first

A probability space \((\Omega,\mathcal F,\mathbf P)\) separates outcomes, measurable events, and their probabilities. A random variable \(X:\Omega\to\mathbf R\) must be measurable.

$$ \mu_X(B)=\mathbf P(X\in B)=\mathbf P(X^{-1}(B)) $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ \mathbf E[g(X)]=\int_{\mathbf R}g(x)\,d\mu_X(x) $$

A small case in full

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \begin{cases}\mu_X(\{k\})=p_k&\Longrightarrow\mathbf E[g(X)]=\sum_kg(k)p_k,\\d\mu_X=f_X(x)\,dx&\Longrightarrow\mathbf E[g(X)]=\int g(x)f_X(x)\,dx.\end{cases} $$

The reusable statement

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad \mu_X(B)=\mathbf P(X\in B)=\mathbf P(X^{-1}(B)),\\[5pt] \mathsf{C}\;&:\quad \mathbf E[g(X)]=\int_{\mathbf R}g(x)\,d\mu_X(x). \end{aligned} $$

A nearby false statement

Conditioning on a probability-zero event cannot be done by naïvely dividing. Conditional densities and regular conditional probabilities require additional structure.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \mathbf E[g(X)]=\int_{\mathbf R}g(x)\,d\mu_X(x) \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Tue 12 July 2011. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.