Suppose we have a equation $$\iint_{-\infty }^{\infty} xy \frac{1}{2\pi \sigma^2}\exp\left(-\frac{(x-\mu_x)^2+(y-\mu_y)^2}{2\sigma^2}\right) dx dy$$

What property of the integral has been used to rephrase the equation like below: $$\int_{-\infty }^{\infty} x \frac{1}{2\pi\sigma^2} \exp\left(-\frac{(x-\mu_x)^2}{2\sigma^2}\right)dx \cdot \int_{-\infty }^{\infty} y \frac{1}{2\pi\sigma^2} \exp\left(-\frac{(y-\mu_y)^2}{2\sigma^2}\right)dy $$

I would like to know the detailed steps between them.

Thank you so much for your help.


1 Answer 1


This is Fubini theorem. You can inverse integrating over $x$ and $y$, then intergating first over $y$ and terms in $x$ are seen as constant and can move out from the integral. $$\int_X \int_Y f(x) g(y) dx dy=\int_Xf(x) \int_Y g(y) dydx = \int_X f(x) dx \int_Y g(y) dy$$


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