E[(X^2+Y^2)/XY] for Geometric(p) R.V.s

[Jonobro]

The Question
Let X and Y be two independent Geometric(p) random variables. Find E[(X^2+Y^2)/XY].

Formulas
Px(k) = py(k) = pq^(k-1)
E(x) = Σx(p(x))

My attempt at a solution
I am really struggling with this question because I want to apply the LOTUS equation but am unsure how to do it for geometric variables. Any help would be appreciated.

 

[mfb]

I didn't test it but it might be easier to find E[(X^2)/(XY)] as intermediate step.

 

[mathman]

If X or Y has a non-zero probability of being 0, that expectation is infinite.

 

[mfb]

Based on the formula in post 1, the distribution starts at 1.

And I checked it, the suggested intermediate step is useful.