Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose we have three random variables $X,Y,Z$.

1) If $X$ and $Y$ are independent, are they still independent given $Z=z$?

2) If $X$ and $Y$ are independent given $Z=z$, are $X$ and $Y$ independent?

If true, please give a proof. If false, please give a counterexample. Thanks a lot for any reply.

share|improve this question
add comment

1 Answer

up vote 2 down vote accepted

1) Suppose that $Z = z$ if and only if $Y = X$. Then $X$ and $Y$ are no longer independent.

2) Suppose that $X$ always equals $Y$ and $X = Y = c$ (for some constant $c$) in the case where $Z = z$. Then $X$ and $Y$ are (vacuously) independent given $Z = z$, but they are not independent in the general case.

share|improve this answer
    
If $X$ always equals $Y$, then knowledge of one determines the other. How are they independent? –  alancalvitti Feb 14 '13 at 5:03
    
They are not. However, they are given that Z = z, as each one can only assume a single value, regardless of what the other value is. –  Joe Z. Feb 14 '13 at 12:25
    
Ah very good, the probability of the product and the factors is 1. –  alancalvitti Feb 14 '13 at 15:17
    
1) A case where this happens is if you define $Z = X - Y$ and $z = 0$. 2) A case where this happens is if you define $X = Y = Z$ and $c = z$. –  Joe Z. Feb 14 '13 at 16:40
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.