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can anyone help with an example for a PMF or some density that:

$P(A,B|C)=P(A|C)P(B|C )$ but $P(A,B)\not=P(A)P(B)$

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1 Answer 1

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Take any two events $A$ and $B$ which are not independent and take $C=A$.

Note that $P(A,B|C)=\frac {P(A\cap B)} {P(A)}$ and $P(A|C)P(B|C)=(1)(\frac {P(A\cap B)} {P(A)}=\frac {P(A\cap B)} {P(A)}$.

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