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

Question

X1=Θ1+W1,X2=Θ1+Θ2+W2.
Find the MAP estimate θ^=(θ^1,θ^2) of (Θ1,Θ2) if we observe that X1=1, X2=3. (You will have to solve a system of two linear equations.)

Answer 1

theta_1=1 theta_2=1