Let the joint distribution of $X$ and $Y$ be given by:

$f(x,y) = e^{-x}$ if $0 < y \leq x < \infty$

Define $Z = X+Y$ and $W = X-Y$

  1. Find the joint pdf of $Z$ and $W$
  2. Calculate $f_{ZW} (0.1,0.5)$


I think I should calculate the pdf of $Z=X+Y$ and then the pdf of $W=X-Y$. I can then multiply the two pdf's if they are independent, i think they are independent but I am not sure.

  • $\begingroup$ "I think I should calculate the pdf of $Z=X+Y$ and then the pdf of $W=X-Y$" This is useless, since you are asked for the joint PDF of $(Z,W)$. The standard change of variables method works like a charm here, starting from $f$ the joint PDF of $(X,Y)$ defined on $\mathbb R^2$ as $$f(x,y)=e^{-x}\mathbf 1_{0<y\leqslant x}.$$ $\endgroup$
    – Did
    May 14 '16 at 5:48

With $X$ and $Y$ two continuous r.v.'s with joint pdf $f_{XY}(x,y)$ and you want to transform to two new variables

$$z=g(x,y) \:\: \text{and} \:\: w=h(x,y) \:\: \:\: \:\:\:\:\:\: (1)$$

then if the transformations is one-to-one and has inverse transformations $x=q(z,w)$ and $y=r(z,w)$ then the joint pdf of $Z$ and $W$ is

$$f_{ZW}(z,w) = f_{XY}(x,y)|J(x,y)|^{-1} \:\:\:\:\:\:\:\:\:(2)$$

where $x=q(z,w)$ and $y=r(z,w)$ and

$$J(x,y)= \begin{vmatrix} \frac{\partial{g}}{\partial{x}} & \frac{\partial{g}}{\partial{y}} \\ \frac{\partial{h}}{\partial{x}} & \frac{\partial{h}}{\partial{y}} \\ \end{vmatrix} = \begin{vmatrix} \frac{\partial{z}}{\partial{x}} & \frac{\partial{z}}{\partial{y}} \\ \frac{\partial{w}}{\partial{x}} & \frac{\partial{w}}{\partial{y}} \\ \end{vmatrix} $$

which is the Jacobian of the transformation $(1)$. If we define

$$\overline{J}(x,y)= \begin{vmatrix} \frac{\partial{q}}{\partial{z}} & \frac{\partial{q}}{\partial{w}} \\ \frac{\partial{r}}{\partial{z}} & \frac{\partial{r}}{\partial{w}} \\ \end{vmatrix} = \begin{vmatrix} \frac{\partial{x}}{\partial{z}} & \frac{\partial{x}}{\partial{w}} \\ \frac{\partial{y}}{\partial{z}} & \frac{\partial{y}}{\partial{w}} \\ \end{vmatrix} $$

then $(2)$ can be expressed as

$$f_{ZW}(z,w) = f_{XY}[q(z,w),r(z,w)]|\overline{J}(x,y)| $$

And it should be straight forward to finish from here.


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