distribution conditional on two variables

Question

Assume we have two random variables $X$ and $Y$. Then, assume that $X|a < X < b$ has density $f^{a,b}(x)$. Then, assume that $g_{1}(y)<g_{2}(y)$ two functions. Is the following true: $X|g_{1}(Y) < X < g_{2}(Y), Y = y_{0}$ has density $f^{g_{1}(y_{0}),g_{2}(y_{0})}(x)$? What if $X$ and $Y$ are independent?

Answer 1

If $X$ and $Y$ are independent then yes as the distribution of $X$ given $g_1(Y)<X<g_2(Y)$ and $Y=y_0$ is the same as the distribution of $X$ given $g_1(y_0)<X<g_2(y_0)$ and $Y=y_0$, which is itself the same as the distribution of $X$ given $g_1(y_0)<X<g_2(y_0)$ by independence.

In general no. You can find easy counter-examples with $Y=X$.