Conditioning on RV

#1
Okay I have been stuck on a homework problem for quite some time and wanted some feedback to see if my methods were correct:

a) Determine P(X > Y) by conditioning on rv X


starting with part a), I simplified to the sum from x=1 to infinity of Fy(x)px(x) then substituting in gives a final solution.
Now for part b), since Y is continuous we must take the integral from inf to -inf of Fx(y)fY(y)dy. I am not sure if that is correct and would like some reassurance because I am confused how this works when X is discrete and Y is continuous.
Finally for c) I am unsure how to start this, any help or guidance would be greatly appreciated!
 
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