AMH - 3__probability.3_func_independent

Definition of Pullbacks

(Informal) Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function. The pullback of \(f\) is the function \(f^{-1}:\mathcal{P}(\mathbb{R}) \rightarrow \mathcal{P}(\mathbb{R})\) defined by \[f^{-1}(A) = \{x \in \mathbb{R} : f(x) \in A\}\]

Example: Let \(f(x) = x^2.\) Then \(f^{-1}(\{4\}) = \{-2,2\}\) and \(f^{-1}((-\infty,0)) = \emptyset.\)

Claim: If \(X\) and \(Y\) are independent random variabes, and \(f\) and \(g\) are functions, then \(f(X)\) and \(g(Y)\) are also independent random variables.

▼ Proof:

Let \(A\) and \(B\) be subsets of \(\mathbb{R}.\) Then \begin{align} P(f(X) \in A, g(Y) \in B) & = P(X \in f^{-1}(A), Y \in g^{-1}(B)) \\ & = P(X \in f^{-1}(A))P(Y \in g^{-1}(B)) \\ & = P(f(X) \in A)P(g(Y) \in B) \end{align} where the second line follows by the independence of \(X\) and \(Y.\)

Example: If \(X\) and \(Y\) are independent, then \(e^X\) and \(e^Y\) are independent. So are \(X\) and \(Y^2.\)