Can the total probability be applied to continuous distributions?
Suppose a random variable X with probability distribution $f_{X4}$. Suppose another random variable Y with probability distribution $f_Y$. Now if we partition $f_X$ into three regions, and obtain $f_Y | f_{X1}, f_Y | f_{X2}$, and $f_Y|f_{X3}$. Can we say:
$f_Y|f_X = (f_Y | f_{X1})P_{X1} + (f_Y | f_{X2})P_{X2} + (f_Y | f_{X3})P_{X3}$
where $P_{X1}$ = integral of $f_{X1}$ with the range of integral equal to limits of $f_{X1}$, and so on.