Random function

Distribution of a differentiable invertible random function

Let be a continuous random variable with probability density function and let be a differentiable and strictly increasing random function. Then #m/thm/prob

where . Note the same applies for a strictly decreasing random function.

Proof

Since , we have

and differentiating both sides gives the above expression.

Equivalently

Multiple dimensions

Let be a random vector with joint probability density function and let be a differentiable and injective Random function. Then #m/thm/prob

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