Let Θ_{1}, Θ_{2}, W_{1}, and W_{2} be independent standard normal random variables. We obtain two observations,

X_{1}=Θ_{1}+W_{1},X_{2}=Θ_{1}+Θ_{2}+W_{2}.

Find the MAP estimate θ^=(θ^_{1},θ^_{2}) of (Θ_{1},Θ_{2}) if we observe that X_{1}=1, X_{2}=3. (You will have to solve a system of two linear equations.)

**θ****^**_{1 = ___????}

**θ^**_{2 =____????}

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