# What's the relationship between statement "Z causes both X and Y" and "X and Y are independent given Z"?

Suppose I have two statements:

Statement 1: Random variable Z is the common cause for random variable X and Y (Z causes both X and Y)

Statement 2: Random variable X and Y are (conditionally) independent given Z.

What's the relationship between Statement 1 and Statement 2? Does Statement 1 imply statement 2, or does statement 2 imply statement 1, or they have other relationships?

Example: Z could be the grade level(or age) for a primary school student, X could be his height, Y could be his math ability. Thanks!

• When you say in Statement 1 that Z is "the" comment cause for X and Y, do you mean that X does not directly cause Y? Oct 30, 2022 at 0:22
• @AdrianKeister Yes, I implicitly assumed this. Oct 30, 2022 at 0:32
• This question seems to be about the difference between causality and statistical associations. There are many related CV threads, incl. Is there an example of two causally dependent events being logically (probabilistically) independent?, Does statistical independence mean lack of causation?, Conditional probability and causality. Oct 30, 2022 at 9:41
• @dipetkov Thanks! I've just read these threads, indeed many insightful comments. Nov 2, 2022 at 21:45
• @dipetkov You have closed this question in favour of stats.stackexchange.com/questions/454516/…, but the previous thread does not even mention conditional independence so I do not see how it can be taken to be a complete answer to the current question, which is specifically about conditional independence. Linking to previous questions about causality was very helpful but closing seems to me to be incorrect. Nov 14, 2022 at 6:58