Lagmental Vicfred

Independence Factorizes Every Joint Event by Vicfred

Last updated: Wed 13 December 2023

tags: mathematics, conditional probability, random variables, distributions

Two random variables are independent when their joint law is the product of the marginal laws. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

The mathematical object

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.

$$ X\perp Y $$

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.

$$ \mathbf P(X\in A,Y\in B)=\mathbf P(X\in A)\mathbf P(Y\in B) $$

One explicit computation

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ f_{X,Y}(x,y)=f_X(x)f_Y(y)\Longrightarrow\mathbf E[g(X)h(Y)]=\mathbf E[g(X)]\mathbf E[h(Y)] $$

Why the identity matters

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad X\perp Y,\\[5pt] \mathsf{C}\;&:\quad \mathbf P(X\in A,Y\in B)=\mathbf P(X\in A)\mathbf P(Y\in B). \end{aligned} $$

Where it can fail

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \mathbf P(X\in A,Y\in B)=\mathbf P(X\in A)\mathbf P(Y\in B) \end{gathered}} $$

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 Fri 12 May 2017. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.