A box contains eight balls numbered from 1 to 8. The first four are red and the other white. We select the balls randomly from the box and define the following variables: X is the number of white balls in the sample, Y the number of even numbers and Z the number of balls in the sample whose number is less than 6. Find the joint distribution of the following random vectors (X, Y); (X, Z); (Y, Z) and (X, Y, Z). Study the independence of these variables.
I'm having trouble trying to define the variables. Here's how I think it should be:
$\Omega= \{1,2,3,4,5,6,7,8\}$ $$X=\{1 \ \text{if} \ x=\{5,6,7,8\}, 0 \ o.c.\}\\ Y=\{1 \ \text{if} \ y=\{2,4,6,8\}, 0 \ o.c.\} \\ Z=\{1 \ \text{if} \ z=\{1,2,3,4,5\}, 0 \ o.c.\} $$