In a paper I encountered the following notation
\begin{equation}P(Z\leq z,u\leq Y\leq v)=C(F_{Z}(z),F_{Y}(v)-F_{Y}(u))\end{equation}
However I don't see why this holds in relation to uniform random variables. Usually $$P(Z\leq,Y\leq v)=P(F_{Z}(Z)\leq F_{Z}(z),F_{Y}(Y)\leq F_{Y}(v))=P(U_{1}\leq F_{Z}(z), U_{2}\leq F_{Y}(v))=C(F_{Z}(z),F_{Y}(v))$$
Can anyone explain to me where the copula $$C(F_{Z}(z),F_{Y}(v)-F_{Y}(u))$$ comes from in terms of uniform random variables?
Usually the copula $C$ represents the distribution function of uniform random variables. But what does the copula above represent? Does this require the reformulation of Sklar's theorem?
I'm really confused and help would be much appreciated.
"$C$" here means cumulative density function. "$P$" means probability density function. The total probability that $Y$ lies between $u$ and $v$ is the difference in the cumulative densities at $v$ and $u$.
Edit: (More on applying Sklar's Theorem and PDF $\leftrightarrow$ CDF...)
The marginal distributions are $F_\gamma(x) = P(\gamma \leq x)$. Sklar's theorem is that $P(X_1 \leq x_1, X_2 \leq x_2, \dots, X_d \leq x_d) = C(F_{X_1}(x_1), \dots F_{X_d}(x_d))$. We find, then \begin{align*} P(Z \leq z, u \leq Y \leq v) &= P(Z \leq z, Y \leq v) - P(Z \leq z, Y \leq u) \\ &= C(F_Z(z), F_{Y}(v)) - C(F_Z(z), F_{Y}(u))\\ &= C(F_Z(z), F_{Y}(v) - F_{Y}(u))\\ \end{align*} We have merely used the fact that the probability that $u \leq Y \leq v$ is the probability that $Y \leq v$ minus the probability that $Y \leq u$, which is the usual way to turn the probability of a variable being in an interval to the difference of the cumulatives at the interval's endpoints.