x_1,x_2,x_3,x_4 each have poisson distribution. x_1 and x_2 are independent , x_1 and x_4 are independent, x_2 and x_3 are independent, x_3 and x_4 are independent.
I also know that x_1+x_2 and x_3 +x_4 are independent (I'm not sure this is relevant in order to prove the following)
I wand to prove that x_1 and x_3 are independent and that x_2 and x_4 are independent.