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.
Asked
Active
Viewed 138 times
2
-
Yes, the link is clarified using measure theory. See here: https://math.stackexchange.com/questions/152338/is-there-a-change-of-variables-formula-for-a-measure-theoretic-integral-that-doe. – Dec 28 '20 at 21:19
-
https://math.stackexchange.com/q/243529/321264, https://math.stackexchange.com/q/3285425/321264 – StubbornAtom Dec 29 '20 at 07:41