## A textbook problem¶

Given \(X\), \(Y\) are two independent random variables, show that functions \(g(X)\) and \(h(Y)\) are independent too

## Short Proof¶

Never took a course on measure theory, so avoiding that route:

Let.

\(R = g(X)\)

\(S = h(Y)\)

Also define,

\(A_r = \{x: g(X)\leq r\}\) and \(B_s = \{y:h(Y) \leq s\}\)

$$
F_{RS}(r,s) = P(R\leq r, S \leq s) = P(X \in A_r, Y \in B_s)
$$

Now, since \(X\), \(Y\) are independent:

$$
P(X \in A_r, Y \in B_s) = P(X \in A_r)P(Y \in B_s) = F_R(r)F_S(s)
$$

Which implies

$$
F_{RS}(r,s) = F_R(r)F_S(s)
$$

and hence

\(g(X), h(Y)\) are independent.

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