Random function

Distribution of a differentiable invertible random function

Let 𝑋 :𝜉 →ℝ be a continuous random variable with probability density function 𝑓𝑋 and let 𝑇 :ℝ →ℝ be a 𝐶1 differentiable and strictly increasing random function. Then prob

𝑓𝑇(𝑡)=𝑓𝑋(𝑥)∣𝑑𝑥𝑑𝑡∣

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

Proof

Since 𝑇(𝑋) ≤𝑡 ⟺ 𝑋 ≤𝑇−1(𝑡), we have

𝐹𝑇(𝑡)=ℙ(𝑇(𝑋)≤𝑡)=ℙ(𝑋≤𝑇−1(𝑡))=𝐹𝑋(𝑇−1(𝑡))

and differentiating both sides gives the above expression.

Equivalently

𝑓𝑇(𝑡)𝑑𝑡=𝑓𝑋(𝑥)𝑑𝑥

Multiple dimensions

Let ⃗𝐗 :𝜉 →ℝ𝑛 be a random vector with joint probability density function 𝑓⃗𝐗 and let ⃗𝐓 :ℝ𝑛 →ℝ𝑚 be a 𝐶1 differentiable and injective Random function. Then prob

𝑓⃗𝐓(⃗𝐭)=𝑓⃗𝐗(⃗𝐱)∣det𝐷⃗𝐓−1∣

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