How is the change of variables formula from multivariable calculus $$\int_{\Omega} f \circ \phi \cdot |\text{Jac}(\phi)| = \int_{\phi(\Omega)} f$$ related to the change of variables formula for expectations? $$E(g(X)) = \int g(x)f(x)\ dx.$$ Can you use the first to derive the second? Is the link between them made clear using measure theory? If $X$ is a continuous random variable on the probability space $(\Omega, \mathcal{F}, P)$, I know that its expectation is given by $$E(X) = \int_{\Omega} X\ dP.$$ But I'm not sure where to go from here.
2026-03-29 16:03:13.1774800193
How are the change of variables formula from calculus and the change of variables formula for expectation related?
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