Multivariate distribution - CDF calculation rules

Question

It is basic knowledge that we can write $P(a \leq x \leq b) = P(x \leq b) - P(x \leq a)$. But what if we have a multivariate distribution?

Here I know $P(Z \leq Z_x, Z<Z_y) = \Phi_2(Z_x,Z_y,\rho)$ where $\rho$ is the correlation parameter and $\Phi_2$ is the bivariate cumulative normal distribution function. So, how can I write: $P(a \leq Z_x \leq b, c \leq Z_y \leq d)$ with the use of $\Phi_2(Z_x,Z_y,\rho)$?

Use inclusion-exclusion – kjetil b halvorsen Apr 23 '18 at 08:13 — kjetil b halvorsen, Apr 23 '18 at 08:13

score 1 · Accepted Answer · edited Apr 23 '18 at 08:39

1

If $X$ and $Y$ have joint distribution then: $$\mathsf P(a<X\leq b,c<Y\leq d)=F_{X,Y}(b,d)-F_{X,Y}(a,d)-F_{X,Y}(b,c)+F_{X,Y}(a,c)$$

edited Apr 23 '18 at 08:39

Did

279,727

answered Apr 23 '18 at 08:26

drhab

151,093

@Did Thank you (sigh.., how could I. I really don't understand myself). – drhab Apr 23 '18 at 08:43
Well, we all have our moments, I guess... :-) – Did Apr 23 '18 at 09:57

Multivariate distribution - CDF calculation rules

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