Conditions for $X$ and $Y$ independent of $Z$

Question

To show that the continuous random variables $X$ and $Y$ with joint PDF $f_{XY}(x,y)$ are independent of a third random variable $Z$ with PDF $f_Z(z)$, is it enough to show that $$f_{XY|Z}(x,y|z)=f_{XY}(x,y), \ \forall x,y,z$$ or do I also need to show that $f_{X|Z}(x|z)=f_X(x) \ \forall x,z$ and $f_{Y|Z}(y|z) \ \forall y,z$?

Answer 1

To show joint independence of $X$ and $Y$ from $Z$, it is enough to show that $$f_{XYZ}(x,y,z)=f_{XY}(x,y)f_Z(z) \ \forall x,y,z$$ From this, \begin{align*} f_{XY|Z}(x,y|z)&=\frac{{f_{XYZ}}\left( {x,y,z} \right)}{f_Z(z)} \\ &=\frac{{f_{XY}}\left( {x,y} \right)f_Z(z)}{f_Z(z)}\\ &={f_{XY}}\left( {x,y} \right) \end{align*} Also \begin{align*} f_{XZ}(x,z)&=\int_{-\infty}^{\infty}f_{XYZ}(x,y,z)dy\\ &=\int_{-\infty}^{\infty}f_{XY}(x,y)f_Z(z)dy\\ &=f_Z(z)\int_{-\infty}^{\infty}f_{XY}(x,y)dy\\ &=f_X(x)f_Z(z) \end{align*} which holds for $Y$ as well.