Distribution of a differentiable invertible random function

Let $𝑋 : 𝜉 \to ℝ$ be a continuous random variable with probability density function $𝑓 𝑋$ and let $𝑇 : ℝ \to ℝ$ be a $𝐶 1$ differentiable and strictly increasing random function. Then #m/thm/prob

𝑓 𝑇 (𝑡) = 𝑓 𝑋 (𝑥) ∣ 𝑑 𝑥 𝑑 𝑡 ∣

where $𝑥 = 𝑇 - 1 (𝑡)$ . Note the same applies for a strictly decreasing random function.

Proof

Since $𝑇 (𝑋) \leq 𝑡 ⟺ 𝑋 \leq 𝑇 - 1 (𝑡)$ , we have

𝐹 𝑇 (𝑡) = ℙ (𝑇 (𝑋) \leq 𝑡) = ℙ (𝑋 \leq 𝑇 - 1 (𝑡)) = 𝐹 𝑋 (𝑇 - 1 (𝑡))

and differentiating both sides gives the above expression.

Equivalently

𝑓 𝑇 (𝑡) 𝑑 𝑡 = 𝑓 𝑋 (𝑥) 𝑑 𝑥

Multiple dimensions

Let $⃗ 𝐗 : 𝜉 \to ℝ 𝑛$ be a random vector with joint probability density function $𝑓 ⃗ 𝐗$ and let $⃗ 𝐓 : ℝ 𝑛 \to ℝ 𝑚$ be a $𝐶 1$ differentiable and injective Random function. Then #m/thm/prob

𝑓 ⃗ 𝐓 (⃗ 𝐭) = 𝑓 ⃗ 𝐗 (⃗ 𝐱) ∣ d e t 𝐷 ⃗ 𝐓 - 1 ∣

#state/tidy | #lang/en | #SemBr