Find the conditional distribution

Question

Suppose that
(i) ε ∼ N (0, σ2);
(ii) X ∼ N (µ, τ 2);
(iii) ε ⊥⊥ X;
(iv) Y = α + β X + ε.
(a) Find the conditional distribution of Y given X = x.

 

the solution is:  Y |X = x ∼ N (α + β x, σ2)

But, to be honest, I do not know how to solve this question.........

 

Answer 1

Plugging in X = x in (iv) gives:

 

Y|(X=x) = α + βx + ε

 

The normal distribution family is closed under linear transformation. That is, any linear function of a normal random variable is itself normal. This and (i) imply the final answer:

 

Y|(X = x) ∼ N(α + βx, σ²)

 

Note that (ii) and (iii) don't affect the answer to the question.