Random independent variables $x_1, x_2 \sim \ \ \text{poisson(}\lambda )$
$y=x_1+x_2$
$z=x_1-x_2$
The possible density function $f(y,z)=?$ by using Inverse transformation method.
Note that I can solve this question by moment generating function
But it's difficult to apply inverse transformation method.
First of all,
$x_1=(y+z)/2 $ and $ x_2=(y-z)/2$
And $$f(x_1, x_2)= \frac{e^{-2\lambda}\lambda^{(x_1+x_2)}}{x_1! x_2!}$$
$$f(y,z)= \sum_{x_2} \sum_{x_1}f(x_1,x_2)=\sum\sum \frac{e^{-2\lambda}\lambda^{y}}{\left(\frac{y+z}{2}\right)! \left(\frac{y-z}{ 2}\right)!}$$
In this point I'm stacking. Thank you for helping.