Expectations and Variance of linear regression model

Question

Consider,

Y = c + βX + ε where E(ε|X) = 0 and Var(ε|X) = (σ sub ε)^2.

Assume Var(X) = (σ sub X)^2. Find Var(E(Y|X)).

So far I have,

(E(Y|X)) = E(E(Y|X)^2) - (E(E(Y|X)))^2

I'm not sure where to go from here.

Answer 1

First $$ E[Y\mid X] = E[c + \beta X + \epsilon\mid X] = E[c\mid X] + \beta E[X\mid X] + E [\epsilon\mid X] \\ = c + \beta X $$ so $$ V[E[Y\mid X]] = V[c + \beta X] = \beta^2 V[X] = \beta^2 σ_X^2 $$