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Please help me formulate a multivariate quantile function to facilitate the random sampling of a distribution. I have an idea, and I want to get feedback if it makes mathematical sense.

Let there be the continuous random variables $X$ and $Y$.

Let us assume we are given the PDFs $f_{X,Y}$, $f_X$, and $f_Y$.

We can then define the conditional CDFs... $$F_{X|Y}(x, y) = \mathbb P(X \le x | Y = y) = {1 \over f_Y(y)} \int_{- \infty}^x f_{X, Y}(t, y) dt$$

$$F_{Y|X}(y, x) = \mathbb P(Y \le y | X = x) = {1 \over f_X(x)} \int_{- \infty}^y f_{X, Y}(x, t) dt$$

(See https://mathcs.clarku.edu/~djoyce/ma217/conddist.pdf for explanation of above formulas)

We can then define the quantile function $\omega_{X, Y}$ in the following way... $$\omega_{X, Y}(p_1, p_2) = (x, y)$$ $$\text{where }p_1, p_2 \in [0, 1]$$ $$\text{such that...}$$ $$F_{X|Y}(x, y) = p_1$$ $$\text{ and }$$ $$F_{Y|X}(y, x) = p_2$$

Is this a mathematically valid way to construct a multivariate quantile function?

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That will not work (unless $X$ and $Y$ are independent) and indeed it may not give you a way into the calculation as you probably need to know $y$ to solve $F_{X\mid Y}(x \mid y) = p_1$ for $x$, but you also need to know $x$ to solve $F_{Y\mid X}(y\mid x) = p_2$ for $y$

But it is not far away. Instead, try solving

$$F_{X}(x) = p_1$$ for $x$, and then use that $x$ to solve $$F_{Y\mid X}(y \mid x) = p_2$$ for $y$

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