Functions of Independent Random Variables are Independent

Theorem

Let $X$ and $Y$ be independent random variables on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $g$ and $h$ be real-valued functions defined on the codomains of $X$ and $Y$ respectively.

Then $\map g X$ and $\map h Y$ are independent random variables.

Proof

Let $A$ and $B$ be subsets of the real numbers $\R$.

Let $g^{-1} \sqbrk A$ and $h^{-1} \sqbrk B$ denote the preimages of $A$ and $B$ under $g$ and $h$ respectively.

Applying the definition of independent random variables:

\(\ds \map \Pr {\map g X \in A, \map h Y \in B}\)

\(=\)

\(\ds \map \Pr {X \in g^{-1} \sqbrk A, Y \in h^{-1} \sqbrk B}\)

Definition of Preimage of Subset under Mapping

\(\ds \)

\(=\)

\(\ds \map \Pr {X \in g^{-1} \sqbrk A} \map \Pr {Y \in h^{-1} \sqbrk B}\)

Definition of Independent Random Variables

\(\ds \)

\(=\)

\(\ds \map \Pr {\map g X \in A} \map \Pr {\map h Y \in B}\)

Definition of Preimage of Subset under Mapping

Hence $\map g X$ and $\map h Y$ are independent random variables.

$\blacksquare$

Sources

• 2005: Neil A. Weiss: A Course in Probability: $\S 6.4$