X and Y are independent implies X and f(Y) are independent

Dependencies:

1. Independence of random variables (incomplete)
2. σ-algebra
3. Measurable function

$\newcommand{\Fcal}{\mathcal{F}}$ $\newcommand{\Dhat}{\widehat{D}}$ $\newcommand{\Shat}{\widehat{S}}$ $\newcommand{\That}{\widehat{T}}$ $\newcommand{\Fcalhat}{\widehat{\mathcal{F}}}$ Let $X_1$ and $X_2$ be random variables over $\sigma$-algebras $(D_1, \Fcal_1)$ and $(D_2, \Fcal_2)$, respectively. Let $(\Dhat, \Fcalhat)$ be a $\sigma$-algebra. Let $f: D_2 \mapsto \Dhat$ be a measurable function from $(D_2, \Fcal_2)$ to $(\Dhat, \Fcalhat)$.

If $X_1$ and $X_2$ are independent, then $X_1$ and $f(X_2)$ are also independent.

Proof

Let $S_1 \in \Fcal_1$ and $\Shat \in \Fcalhat$. We will prove that $\Pr(X_1 \in S_1 \cap f(X_2) \in \Shat) = \Pr(X_1 \in S_1)\Pr(f(X_2) \in \Shat)$. This would imply that $X_1$ and $f(X_2)$ are independent.

For any set $\That \subseteq \Dhat$, define $f^{-1}(\That) = \{x_2 \in D_2: f(x_2) \in \That\}$. Therefore, $x_2 \in f^{-1}(\That) \iff f(x_2) \in \That$. Since $f$ is measurable, we get that $\That \in \Fcalhat \implies f^{-1}(\That) \in \Fcal_2$. Let $S_2 = f^{-1}(\Shat)$. Therefore, $S_2 \in \Fcal_2$.

\begin{align} & \Pr(X_1 \in S_1 \cap f(X_2) \in \Shat) \\ &= \Pr(X_1 \in S_1 \cap X_2 \in S_2) \\ &= \Pr(X_1 \in S_1)\Pr(X_2 \in S_2) \tag{since $X_1$ and $X_2$ are independent} \\ &= \Pr(X_1 \in S_1)\Pr(f(X_2) \in \Shat) \end{align}

Dependency for: None

Info:

• Depth: 8
• Number of transitive dependencies: 16

Transitive dependencies:

1. /analysis/topological-space
2. /sets-and-relations/countable-set
3. /sets-and-relations/de-morgan-laws
4. σ-algebra
5. Generated σ-algebra
6. Borel algebra
7. Measurable function
8. Generators of the real Borel algebra (incomplete)
9. Measure
10. σ-algebra is closed under countable intersections
11. Probability
12. Conditional probability (incomplete)
13. Independence of events
14. Independence of composite events
15. Random variable
16. Independence of random variables (incomplete)