Question

Let X, Y, and Z be random variables such that X and Y are conditionally independent given Z. Which of the following statements is false?

a. P(X|Y,Z)=P(X|Z)
b. P(X,Z|Y)=P(X,Z)
c. P(X,Y|Z)=P(X|Z)P(Y|Z)
d. P(X,Y,Z)=P(X|Z)P(Y|Z)P(Z)

Answer 1

The correct answer is b. P(X,Z|Y)=P(X,Z).

Explanation:

a. P(X|Y,Z)=P(X|Z) is true. This follows from the definition of conditional independence: if X and Y are independent given Z, then the probability of X given both Y and Z is the same as the probability of X given Z alone.

c. P(X,Y|Z)=P(X|Z)P(Y|Z) is also true. This follows from the definition of conditional independence: if X and Y are independent given Z, then the probability of both X and Y given Z is the product of the probabilities of X and Y given Z separately.

d. P(X,Y,Z)=P(X|Z)P(Y|Z)P(Z) is also true. This is the definition of the joint probability of three variables.

Therefore, the false statement is b. P(X,Z|Y)=P(X,Z). In general, P(X,Z|Y) is not equal to P(X,Z). The probability P(X,Z|Y) represents the probability of X and Z occurring given that Y has occurred, while P(X,Z) represents the probability of X and Z occurring independently of each other.