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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.

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1 Answer

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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.

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If $X$ always equals $Y$, then knowledge of one determines the other. How are they independent? – alancalvitti Feb 14 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 at 12:25
Ah very good, the probability of the product and the factors is 1. – alancalvitti Feb 14 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 at 16:40

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