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In single dimension, the probability that a random variable $X$ is greater than some value $x$ is easily related to the cumulative distribution(c.d.f.) as $Pr(X > x) = 1 - F(x)$ if only $Pr[X \leq x] = F(x)$ is known.

If instead we have a random vector $[X_1, X_2, \cdots, X_n]$, could you please let me know if there is a direct relation between the c.d.f and the probability $Pr[X_1 > x_1, X_2 > x_2, \cdots, X_n > x_n]$ if I only know $F(x_1, x_2, \cdots, x_n) = Pr[X_1 \leq x_1, X_2 \leq x_2, \cdots, X_n \leq x_n]$? I know it is not $1 - F[x_1, x_2, \cdots, x_n]$ in this case.

randomprime
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    $\newcommand{P}{\Bbb{P}}$Hint: Let $A_i = { X_i> x_i}$ and $B_i = A_i^c = { X_i \le x_i}$. Note that $$\begin{align}\P\left(\bigcap_{i=1}^{n} A_i\right) &= \P\left(\bigcap_{i=1}^{n} B_i^c \right) \ &= \P\left(\left(\bigcup_{i=1}^{n} B_i \right)^c\right) \ &= 1- \P\left(\bigcup_{i=1}^{n} B_i \right). \end{align}$$

    Now, the probability of this union can be obtained using the inclusion-exclusion principle. It will involve probabilities of intersections of the $B_i$, which are obtainable from the joint CDF $F$ by taking limits to $\infty$ of the other variables.

     – Minus One-Twelfth May 10 '19 at 20:39
  • Closely related: https://stats.stackexchange.com/questions/388713. Likely many answers can be found in the hits at https://stats.stackexchange.com/search?q=copula+cdf, too. – whuber May 10 '19 at 21:44

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