i am faced with the following problem which i can't seem to solve…
Let $X_1$ and $X_2$ two independent normal distributions of parameters $\mu$ and $\sigma^2$ and let $Y_1 = X_1 + X_2$ and $Y_2 = X_1 + 2X_2$.
Compute the joint probability distribution of $Y_1$ and $Y_2$
This is what i tried
$$f_{Y_2\vert Y_1}(y_2)f_{Y_1}(y_1) = f_{Y_1,Y_2}(y_1, y_2)$$
For a given $x_1 + x_2 = y_1$, we have $x_1 + 2x_2 = y_2 \iff x_2 = y_2 – y_1$
Hence
$$f_{Y_2\vert Y_1}(y_2) = f_{X_2}(y_2 – y_1)$$
$$f_{Y_1,Y_2}(y_1, y_2) = f_{X_2}(y_2 – y_1)f_{Y_1}(y_1) $$
$Y_1 = X_1 + X_2 \implies Y_1$ is normal of parameters $2\mu$ and $2\sigma^2$ and $X_2$ is normal of parameters $\mu$ and $\sigma^2$ so we can easily compute the product.
However, with this approach, i do not get the desired result…
Could anyone see where this reasoning fails?
Thank you in advance.
Best Answer
The Jacobian transformation you attempted should go:
$\left\{\raise{1ex}{Y_1=X_1+X_2\\Y_2=X_1+2X_2}\right\} \iff\left\{\raise{1ex}{X_1=2Y_1-Y_2\\X_2=Y_2-Y_1}\right\}$
So therefore $f_{Y_1,Y_2}(u, v) ~=~ \begin{Vmatrix}\dfrac{\partial (2u-v,v-u)}{\partial (u,v)}\end{Vmatrix} f_{X_1}(2u-v)f_{X_2}(v-u) $