Does the Expectation of the random variable $X^{-1}Y^{-1}Z$require that we know the joint density of $X^{-1}Y^{-1}Z$?
Solution 1:
By the Law Of The Unconcious Statistician (LOTUS), knowing the joint density of X,Y and Z will be enough. In fact, note that from the joint density of X,Y and Z you can compute the density of the random variable $X^{-1}Y^{-1}Z$ (This gives you a hint of why LOTUS is true). Sometimes you can derive densities of transformations of random variables from first principles. For the general case, see proposition 1.8 in Shao, Mathematical Statistics:
