# Formula for the multivariate cumulative distribution function (continuous case)

Consider a random vector of dimension $$5\times 1$$, $$X\equiv (X_1, X_2, X_3, X_4, X_5)$$, with joint CDF denoted by $$G$$.

Consider the following probability $$p\equiv Pr(X_1\geq a_1,X_2\geq a_2,X_3\geq a_3,X_4\geq a_4,X_5\geq a_5 ),$$ where $$(a_1,_2,a_3,a_4,a_5)\in \mathbb{R}^5$$.

I would like to write $$p$$ using the CDF of $$X$$, ($$G$$ introduced above). Could you advise on how to do that?

My thoughts: At the moment, I have been able to derive the formula above for a vector with at most 3 components. For example, when $$X$$ is scalar $$Pr(X_1\geq a_1)=P_{X_1}([a_1, \infty])=1-G(a_1)$$

When $$X$$ is bivariate $$Pr(X_1\geq a_1, X_2\geq a_2)=P_{X_1,X_2}([a_1, \infty]\times [a_2, \infty])=1+G(a_1,a_2)-G(\infty, a_2)-G(a_1, \infty)$$ (see also this question).

When $$X$$ is trivariate $$Pr(X_1\geq a_1, X_2\geq a_2, X_3\geq a_3)=P_{X_1,X_2,X_3}([a_1, \infty]\times [a_2, \infty]\times [a_3, \infty])=1-G(a_1,a_2,a_3)+G(\infty, a_2, a_3)+G(a_1, \infty, a_3)-G(\infty, \infty, a_3)+G(a_1, a_2, \infty)-G(\infty, a_2, \infty)-G(a_1, \infty, \infty)$$ How do I extend this to the $$5$$-variate case?

• Are you speaking about discrete or continuous case? Jul 24, 2020 at 18:21
• Continuous case
– TEX
Jul 24, 2020 at 18:21
• If we know (exists) density function, then multivariate integral should give answer. Jul 24, 2020 at 18:25
• Thanks, we know that there exists a density function. I've been trying to compute the integral but it becomes messy. Could you help me with an answer?
– TEX
Jul 24, 2020 at 18:27

Let's denote $$\mathcal{P}(a_1,\cdots,a_n) = Pr(X_1\geqslant a_1, X_2\geqslant a_2, \cdots, X_n\geqslant a_n) =\\= \int\limits_{a_1}^{\infty}\int\limits_{a_2}^{\infty} \cdots \int\limits_{a_n}^{\infty}fdx_1dx_2\cdots dx_n$$ And known cumulativie one with $$\mathcal{G}(a_1,\cdots,a_n) = Pr(X_1\leqslant a_1, X_2\leqslant a_2, \cdots, X_n\leqslant a_n) =\\= \int\limits_{-\infty}^{a_1}\int\limits_{-\infty}^{a_2} \cdots \int\limits_{-\infty}^{a_n}fdx_1dx_2\cdots dx_n$$ Then $$1-\mathcal{P}(a_1,\cdots,a_n) = \sum\limits_{b_{i_k} \ne \infty}\mathcal{G}(b_{i_1},\cdots,b_{i_n}) - \sum\limits_{ b_{i_k} \ne \infty, b_{i_j} \ne \infty}\mathcal{G}(b_{i_1},\cdots,b_{i_n}) +\\+ \sum\limits_{ b_{i_k} \ne \infty, b_{i_j} \ne \infty \\b_{i_l} \ne \infty }\mathcal{G}(b_{i_1},\cdots,b_{i_n}) - \cdots +(-1)^{n-1}\mathcal{G}(a_1,\cdots,a_n)$$ Where in first sum all arguments are $$\infty$$ except only one, which equals some $$a_i$$, so we have $$n$$ such members. In second sum all arguments are $$\infty$$ except two ones, so we have all members with only 2 $$a_i$$ arguments, so $$\frac{n(n-1)}{2}$$ members and so on.