Multivariate normal distribution

Question

1. Suppose that (i) ε ∼ N(0,σ2); (ii) X∼N(μ,τ2); (iii) ε⊥X; (iv) Y =α+βX+ε.
(a) Find the conditional distribution of Y given X = x.

This is the answer: (a) Y|X=x∼N(α+βx,σ2)
But I have no idea how to get this. 

Answer 1

I will rewrite the conditions in clearer formatting:

 

(i) ε ∼ N(0 , σ²)

(ii) X ∼ N(μ , τ²)

(iii) ε ⊥ X

(iv) Y = α + βX + ε

 

Solution:

 

Using condition 4 we get:

 

E(Y|X) = E(α + βX + ε|X)

 

       = α + E(βX|X) + E(ε)    ← Linearity of expectation

 

       = α + βX + E(ε)            ← Taking out what's known

 

       = α + βX + ε

 

Condition 3 says that the residual ε has the same distribution independent of X.

 

Finally we have

 

Y|X = E(Y|X) + ε ~ E(Y|X) + N(0 , σ²) = N(α + βX + ε , σ²)