Calculating the mean and variance of random variables

Question

I've been trying to solve the question below, but I am very confused. If anyone could lend a hand in helping me understand this problem by studying their solution, it would be much appreciated. Thank you.

Let X and Y be two possibly related/dependent random variables. Express the mean and the variance of X in terms of conditional means and conditional variances of X|Y. I.e., find the functions f(.) and g(.) such that

1. $$​E[X] = f( E[X|Y] )$$

2. $$V[X] = g( E[X|Y], V[X|Y] )$$

Answer 1

1. By the law of total expectation,

$$E[X] = E[E[X|Y]]$$

$f(X)$ then is $E[X]$

2. By the law of total variance,

$$Var[X] = E[Var[X|Y]] + Var[E[X|Y]]$$

$g(X, Y)$ then is $E[Y] + Var[X]$