(Non-)Multiplicativity of the expected value

If two random variables $X$ and $Y$ are independent, the expected value is multiplicative, i.e. $$\mathrm{E}(X\,Y) = \mathrm{E}(X) \, \mathrm{E}(Y)$$

If two random variables $X$ and $Y$ are dependent, the expected value is not necessarily multiplicative, i.e. there exist $X$ and $Y$ such that $$\mathrm{E}(X\,Y) \neq \mathrm{E}(X) \, \mathrm{E}(Y)$$

Compare: https://en.wikipedia.org/wiki/Distribution_of_the_product_of_two_random_variables

References:

1. https://statproofbook.github.io/P/mean-mult.html