Let Θ1, Θ2, W1, and W2 be independent standard normal random variables. We obtain two observations,

Question

Let Θ1, Θ2, W1, and W2 be independent standard normal random variables. We obtain two observations,
X1=Θ1+W1,X2=Θ1+Θ2+W2.
Find the MAP estimate θ^(theta hat)=(θ^1,θ^2) of (Θ1,Θ2) if we observe that X1=1, X2=3.
1.θ^1 (theta hat)=?
2.θ^2(theta hat)=?