How do I find if 2 random variables are conditionally independent given the joint conditional PMF?

Question

This is the question, but I can not find a way on how to tackle it...

I know that if they are independent then:

p(X, Y|Z) = p(X|Z)p(Y|Z)
p(X|Y, Z) = p(X|Z)
p(Y|X, Z) = p(Y|Z)

Does someone have some tips on how to tackle this?

Answer 1

You are to check if P(x=a,y=b|z=c)=P(x=a|z=c)P(y=b|z=c) for every combination of a,b,c equal to 0 or 1. For example, P(x=1,y=1|z=0)=0.49 but also P(x=1|z=0)P(y=1|z=0)=(P(x=1,y=0|z=0)+P(x=1,y=1|z=0))*(P(x=0,y=1|z=0)+P(x=1,y=1|z=0))=(0.49+0.21)(0.49+0.21)=0.49.

There are seven cases to go but once you found that P(x=a,y=b|z=c)\not=P(x=a|z=c)P(y=b|z=c) for some a,b,c you are done, in which case the answer is NO (x,y are not conditionally independent w.r.t. z). Otherwise, the answer is YES.

All the required values can be read from the table.