Probability Theory Integration by substitution

• $(\Omega, \mathcal{F}, P)$,
• $d \in \mathbb{N}$,
• $A, B \subseteq \mathbb{R}^{d}$,
  • open
• $T: A \rightarrow B$
  • injection
  • $C^{1}$
  • Jacobian $J_{T}(x) \neq 0 \ (x \in A)$
• $Y:\Omega \rightarrow \mathbb{R}^{d}$,
  • r.v.
• $p_{Y}$,
  • p.d.f of $Y$
• $X:\Omega \rightarrow \mathbb{R}^{d}$,
  • r.v.
• $p_{X}$,
  • p.d.f of $Y$
  • p.d.f of $X$
• $X = T^{-1}(Y) \ \text{a.s.}$,
  • r.v.

\[p_{X}(x) = p_{Y}(T(x)) |J_{T}(x)|\]