Let X, Y, and Z be jointly continuous random variables. Assume that all conditional PDFs and expectations are well defined. E. g., when conditioning on X = x, assume that x is such that fX(x) > 0.
For each one of the following formulas, state whether it is true for all choices of the function g or false (i. e., not true for all choices of g ).
1) E[g(Y)|X=x] = ∫ g(y) fY|X(y|x) dy
2) E[g(y)|X=x] = ∫ g(y) fY|X(y|x) dy
3) E[g(Y)] = ∫ E[g(Y)|Z=z] fz(z) dz