Let X and Y be two random variables Show that VY EVYX VEYX

Let X and Y be two random variables. Show that V(Y) = E[V(Y|X)] + V[E(Y|X)].

Solution

this can be proved by using the law of total expectation which states that

E(X) = E(E(X|Y))

proof: from the definition of variance V(Y) = E[Y²] - E[Y]² now applying the above law

V(Y) = E [ E [ Y²|X ] ] - [ E [ E [ Y|X ] ] ]²  

= E [ V [ Y|X ] ] + [ E [ Y|X ]² ] - [ E [ E [ Y|X ] ] ]²

= E [ V [ Y|X ] ] + E [ Y|X ]² - [ E [ E [ Y|X ] ] ]²

= E [ V [ Y|X ] ] + ( E [ E [ Y|X ] ]² - [ E [ E [ Y|X ] ] ]² ) since the expectation of a sum is the sum of expectations

=  E [ V [ Y|X ] ] + V [ E [ Y|X ] ]