is on the space D$_{0}$ ( W ) by ( ∂ $^{d$_{x}$}$C ( ..., . ) , ψ ) : In the y argument the conditional distribution is an ordinary function so here y is considered just as a parameter of the generalized function. However, the definition of F$_{y}$$_{|}$$_{x}$ in (11) can be extended to a functional for functions defined on the product space; for any ψ$_{x,y}$ = ψ$_{x}$ ( x$_{1}$, ...x$_{d}$$_{x}$ ) ψ$_{y}$ ( y$_{1}$, ..., y$_{d}$$_{y}$ ) ∈ D ((0 , 1) $^{d$_{x}$}$) × D ( R $^{d$_{y}$}$) define the value of the functional by ( F$_{y}$$_{|}$$_{x}$, ψ$_{x,y}$ ) = To define conditional density f$_{y}$$_{|}$$_{x}$ as a generalized function one would have In general, the conditional distribution and conditional density depend on the conditioning variables, x, via the marginals, F$_{x}$ ; considering generalized functions makes this explicit. There are cases when the conditional distribution and conditional density are defined on the Euclidean space R $^{d$_{x}$}$. This is possible if the distribution function F$_{x}$ is strictly monotone in each argument; then the corresponding generalized density function is positive, moreover, since a monotone function 19 ( f$_{y}$$_{|}$$_{x}$, ψ$_{x,y}$ ) = ( − 1) d$_{x}$ + d$_{y}$ ∫ ... ∫ F$_{x,y}$ ( x, y ) ∂ $^{d$_{x}$}$ψ$_{x}$ ( F$_{x}$$_{1}$, ...F$_{x}$$_{d}$$_{x}$ ) ∂ $^{d$_{y}$}$ψ$_{y}$ ( y$_{1}$, ..., y$_{d}$$_{y}$ ) dF$_{x}$$_{1}$ ( x$_{1}$ ) ...dF$_{x}$$_{d}$$_{x}$ ( x$_{d}$$_{x}$ ) dy$_{1}$...dy$_{d}$$_{y}$. (14) ( − 1) d$_{x}$ ∫ ... ∫ F ( x, y ) ∂ $^{d$_{x}$}$ψ$_{x}$ ( F$_{x}$$_{1}$, ...F$_{x}$$_{d}$$_{x}$ ) ψ$_{y}$ ( y$_{1}$, ..., y$_{d}$$_{y}$ ) dF$_{x}$$_{1}$...dF$_{x}$$_{d}$$_{x}$ dy$_{1}$...dy$_{d}$$_{y}$. ( F$_{y}$$_{|}$$_{x}$, ψ ) = ∫ ... ∫ ∂ $^{d$_{x}$}$C ( F$_{x}$$_{1}$, ...F$_{x}$$_{d}$$_{x}$ , F$_{y}$ ) ψ ( F$_{x}$$_{1}$, ...F$_{x}$$_{d}$$_{x}$ ) dF$_{x}$$_{1}$...dF$_{x}$$_{d}$$_{x}$ . (13)