Is there a Bayes rule for when some of the random variables are discrete and continuous? Let $X$ be a continuous random variable, and $Y$ be a discrete random variable
Is there a "Bayes rule" for: 
$P[Y | X = x]$
where $P$ is the probability mass function of $Y$?
Any reference help!
 A: There's also another identity in the case when $\ X\ $ has a density function  $\ \phi\ $ with respect to some measure $\ \mu\ $ on its codomain.  When this is the case $\ Y, X\ $ will also have a joint mass-density function $\ \psi\ $ satisfying the equation  $\ P\left((Y=y)\, \land \left(X\in B\right)\right)=\int_B\ \psi\left(y,x\right)d\mu\left(x\right)\ $, and we then have
$$ P\left(Y=y\,\vert\, X=x\,\right)= \frac{\psi\left(y,x\right)}{\phi\left(x\right)}\ $$
for all $\ y\ $ and $\mu$-almost all $\ x\ $.
A: The proper statement of Bayes law for this case would be that:
Given any continuous variate $X$ with range $R_X$ and discrete variate $Y$ and some joint distribution of $X$ and $Y$ (they need not be independent) then for any set $B \subset R_X$
such that $P (X \in B) > 0$ and any value $y$,
$$
P( Y = y | X\in B )= \frac{P( (Y = y) \wedge (X\in B) )}{P(X\in B)}
$$ 
You can also go the other way, provided that $y$ is in the range of $Y$ with non-zero probability:
$$
P(X\in B | Y = y )= \frac{P( (Y = y) \wedge (X\in B) )}{P(Y = y)}
$$ 
