Conditional Independence and Independence question

akolman

New member
Hello, I am stuck with the following question.

1. Suppose T ind. C |Z, does it follow that T ind. C ?

2. Suppose T ind. C , does it follow that T ind. C |Z?

I think both don't follow, but I don't know how to show it

CaptainBlack

Well-known member
Hello, I am stuck with the following question.

1. Suppose T ind. C |Z, does it follow that T ind. C ?

2. Suppose T ind. C , does it follow that T ind. C |Z?

I think both don't follow, but I don't know how to show it

$$T$$ and $$C$$ independent given $$Z$$ means:
$$P(T \wedge C|Z)=P(T|Z)P(C|Z)$$
Now we are free to define any relation we want between $$T$$ and $$C$$ if $$\neg Z$$ is the case so that
$$P(T \wedge C) \ne P(T)P(C)$$