# Derivation of Sklar's theorem for copula

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?

• 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 $C(u,v)=P(U\leq u, V\leq v) = F(x,y)=P(X\leq x, Y\leq y)$? Aug 30, 2020 at 17:14

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}

• thanks. could you show the differentiation of $F(x,y) = P(X\leq x, Y\leq y)$ and how it resolves to $F(x,y) = c(u,v) \cdot F(x) F(y)$ Aug 29, 2020 at 22:45
• @develarist I've added the steps. Aug 29, 2020 at 22:56
• 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). Aug 30, 2020 at 3:26
• how would there not be densities for random variables? Sep 2, 2020 at 0:33
• Should there be a d^2 in the second to last equation? ie: (d^2*C(u,v) / dudv) Feb 14 at 1:12