# p.d.f. of a linear combination of random vector - Maximum Likelihood Estimation?

I am working on an assignment problem from the Linear Models course. It appears to me that the Maximum Likelihood estimators are involved, but I can'f figure it out. Any help will be much appreciated. Thanks

The random vector $\mathbf y =(y_1, y_2, y_3)^T$ has p.d.f. $f(\mathbf y) = 4(2\pi\sigma^2)^{-\frac 32} exp(-Q/2\sigma^2)$, where

$Q = 3y_1^2 + 3y_2^2 + 3y_3^2 – 2 y_1y_2 – 2y_1y_3 – 2y_2y_3 – 2 \mu (y_1+y_2+y_3) + 3\mu^2$

Find the distribution of $\mathbf w =(w_1, w_2, w_3)^T$ where

$$w_1 = y_1 + y_2 + y_3$$ $$w_2 = -y_1 + y_2$$ $$w_3 = -y_1 – y_2 + 2y_3$$

Show that $w_1, w_2$ and $w_3$ are independent and that only the distribution of $w_1$ contains information about $\mu$.

• No Maximum Likelihood Estimation here. – Did Apr 15 '13 at 7:42

Hint: $3Q=(w_1-3\mu)^2+6w_2^2+2w_3^2.$