Consider three random variables X, Y, and Z, associated with the same experiment. The random variable X is geometric with parameter p∈(0,1). If X is even, then Y and Z are equal to zero. If X is odd, (Y,Z) is uniformly distributed on the set S={(0,0),(0,2),(2,0),(2,2)}. The figure below shows all the possible values for the triple (X,Y,Z) that have X≤8. (Note that the X axis starts at 1 and that a complete figure would extend indefinitely to the right.)

Question

Answer the following with “Yes" or “No":

a) Are Y and Z independent?
- unanswered

b)Given that Z=2, are X and Y independent?
- unanswered

c) Given that Z=0, are X and Y independent?
- unanswered

d) Given that Z=2, are X and Z independent?

Answer 1

a) no
b) yes
c) no
d) yes

Answer 2

Why yes for the last one?