1. Suppose three random variables X , Y , Z have a joint distribution

Question

PX,Y,Z(x,y,z)=PX(x)PZ∣X(z∣x)PY∣Z(y∣z).
Then, are X and Y independent given Z?

2.Suppose random variables X and Y are independent given Z , then the joint distribution must be of the form
PX,Y,Z(x,y,z)=h(x,z)g(y,z),
where h,g are some functions.
(True or False)

Answer 1

1. No
2. True