Two random variables are dependent given a third random variable, and independent otherwise Suppose that $X$ and $Y$ are dependent given $Z$, and independent when not given $Z$. Does this mean that:
$$
p(x,y) = p(x) \cdot p(y) \\
p(x,y|z) \neq p(x|z) \cdot p(y|z)
$$
Also, are there any real-world examples of this scenario?
 A: There will be many examples of the form $Z = f(X, Y)$, where $X$ and $Y$ are two independent random variables and $f$ is some function.
For example, $X$ is the number I roll on a fair die, $Y$ is the number I roll on a second fair die, and $Z$ is $X + Y$.
A: Yes the translation in formula is correct, up to a loose formalism.
A generic "real-world" example can be obtained when $X$ and $Y$ can
have either positive association or a negative association depending
on $Z$. Then $X$ and $Y$ may eventually be independent.
For instance $X$ be the income and $Y$ be the experience and $Z$ be a
type of professional activity. Then $X$ and $Y$ may be positively
associated for some values of $Z$ but negatively associated for some
others e.g. due to technical obsolescence. Examples can occur in
environmetrics when the relation between $X$ and $Y$ is subject to an
"inversion": $Z$ can be the time within day, day in year, rising/falling tide...
Note that this scenario can not happen when the joint distribution of
$X$, $Y$ and $Z$ is normal because the covariance of $X$ and $Y$
conditional on $Z = z$ does not depend on the value $z$. On the
contrary, easy examples are found with $Z$ being discrete or
multimodal.
