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The joint probability distribution, or bivariate CDF, of two random variables $X$ and $Y$ is

$$F(x,y) = P(X\leq x, Y\leq y)$$

where $F(x) = P(X\leq x)$ would be the marginal distribution of $X$.

Sklar's theorem states that the joint distribution can also be represented as the product of the copula and the marginals,

$$F(x,y) = c(u,v) \cdot F(x) F(y)$$

where $u$ and $v$ are the uniform-transformations of marginal distributions $F(x)$ and $F(y)$, respectively, while $c(u,v)$ is the copula of $X$ and $Y$.

How do you move from the first formula to the second?

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  • $\begingroup$ yeah, surprised it was done without integrals and in terms of $F$s and not the underlying $P()$s. is there a source that does this, but while staying in probabilities and differentiating in probabilities, given that $C(u,v)=P(U\leq u, V\leq v) = F(x,y)=P(X\leq x, Y\leq y)$? $\endgroup$
    – develarist
    Aug 30, 2020 at 17:14

1 Answer 1

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Sklar's theorem states that $$F(x,y)=C(F_X(x),F_Y(y))=C(u,v)$$ or, equivalently ($u=F_X(x),v=F_Y(y)$) $$f(x,y)=c(u,v)f_X(x)f_Y(y)$$ where the lowercase $c$ is the density of the copula $C$ (not the copula itself). If you differentiate the first equation wrt $x,y$ and apply the rules of calculus, you can obtain the second one.

And, the first equation is straightforward to prove: $$\begin{align}C(u,v)&=P(U\leq u, V\leq v)=P(F_X(X)\leq F_X(x), F_Y(Y)\leq F_Y(y))\\&=P(X\leq F_X^{-1}(F_X(x)), Y\leq F_Y^{-1}(F_Y(y)))\\&=P(X\leq x, Y\leq y)\\&=F(x,y)\end{align}$$

The density version can be found as below. First, take the derivative of both sides wrt $x$:

$$\frac{\partial F(x,y)}{\partial x}=\frac{\partial C (u,v)}{\partial u}\frac{\partial u}{\partial x}=\frac{\partial C (u,v)}{\partial u}f_X(x)$$ Then, wrt $y$: $$\begin{align}f(x,y)&=\frac{\partial }{\partial y}\left(\frac{\partial C(u,v)}{\partial u}f_X(x)\right)=\frac{\partial }{\partial y}\left(\frac{\partial C(u,v)}{\partial u}\right)f_X(x)\\&=\frac{\partial^2 C(u,v)}{\partial u\partial v}\frac{\partial v}{\partial y}f_X(x)\\&=c(u,v)f_Y(y)f_X(x)\end{align}$$

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  • $\begingroup$ thanks. could you show the differentiation of $F(x,y) = P(X\leq x, Y\leq y)$ and how it resolves to $F(x,y) = c(u,v) \cdot F(x) F(y)$ $\endgroup$
    – develarist
    Aug 29, 2020 at 22:45
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    $\begingroup$ @develarist I've added the steps. $\endgroup$
    – gunes
    Aug 29, 2020 at 22:56
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    $\begingroup$ Maybe also note that Sklar's Theorem is still true when there aren't densities, even though it's harder to prove (and the density product form obviously doesn't exist). $\endgroup$ Aug 30, 2020 at 3:26
  • $\begingroup$ how would there not be densities for random variables? $\endgroup$
    – develarist
    Sep 2, 2020 at 0:33
  • $\begingroup$ Should there be a d^2 in the second to last equation? ie: (d^2*C(u,v) / dudv) $\endgroup$ Feb 14 at 1:12

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