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Given mutual independence of 3 r.v.s X, Y, Z, can we conclude that X and Y are independent, given Z?

Note that I am interested in case when all 3 r.v.s are mutually independent, not only pair X, Y.

In other words, is it true that:

$p_{X,Y,Z}(x,y,z) = p_X(x) \cdot p_Y(y) \cdot p_Z(z) \implies p_{X,Y|Z}(x,y|z) = p_{X|Z}(x|z) \cdot p_{Y|Z}(y|z)$

If this is not true in general, can someone give me an example when this does not hold?

Eugene Loy
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1 Answers1

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Yes it does. If $X,Y,Z$ are mutually independent then you can say

  • $p_{X,Y}(x,y) = p_X(x) \cdot p_Y(y)$
  • $p_{X,Y \mid Z}(x,y\mid z) = p_{X,Y}(x,y)$
  • $p_{X\mid Z}(x\mid z) = p_X(x)$
  • $p_{Y\mid Z}(y\mid z) = p_Y(y)$

and so $$p_{X,Y\mid Z}(x,y\mid z) =p_{X,Y}(x,y)= p_X(x) \cdot p_Y(y)= p_{X\mid Z}(x\mid z) \cdot p_{Y\mid Z}(y\mid z)$$

It is not necessarily true if $X,Y,Z$ are only pairwise independent.

Henry
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