Let's consider a absolutely continuous random vector $V \equiv (X,Y)$ and its associated joint distribution function $F(x,y)=Pr(X<=x,Y<=y) = \int_{-\infty}^{x}\int_{-\infty}^{y} f(x,y)dxdy$.
If we take four points in the xy plane that are vertexes of a rectangle $R$, $A \equiv (x_1,y_1) \;, B \equiv (x_1,y_0) \;,C \equiv (x_0,y_0) \;,A \equiv (x_0,y_1)$, with $x_1 > x_0$ and $y_1>y_0$, it is well known that the probability that the values of the random vector $V$ are within the rectangle $R$ is given by the value of the distribution function $F(x,y)$ taken at those points according to the below formula:
$Pr(x_0
Is there an explicit formula, generalizing the one above, that applies when we move from 2 to N dimensions ?
In other words, given the distribution function $F(x_1,...,x_N)=Pr(X1<=x1,..,X_N<=x_N)$, it is there a formula that allows to compute $Pr(a_1
What happens if values $a_i$ or $b_j$ are allowed to be $+\infty$ ?
In two dimensions we find the value of the 1-dimensional marginal distributions $F_i(x)$, what's found in the N-dimensional case ?
From computational point of view, is this formula applicable in practice for value of N equal to 10 ? I suppose it involves $2^{10}$ vertexes ...