I ran across this identity in some actuarial literature:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]Pr( (x_1 \le X \le x_2) \ \cap \ (y_1 \le Y \le y_2) ) = F(x_2, y_2) - F(x_1, y_2) - F(x_2, y_1) + F(x_1, y_1)[/tex]

First of all, I am not certain this is correct. I think the expression on the LHS is equal to the following double integral, which is by no means obviously equal to the CDF expression on the RHS:

[tex]Pr( (x_1 \le X \le x_2) \cap (y_1 \le Y \le y_2) ) = \int_{x_1 }^{x_2}\int_{y_1}^{y_2}f(x,y)dydx[/tex]

I suspect that maybe the author intended to use the OR condition in the expression on the left. Did he mean to say this?

[tex]Pr( (x_1 \le X \le x_2) \ \cup \ (y_1 \le Y \le y_2) ) = F(x_2, y_2) - F(x_1, y_2) - F(x_2, y_1) + F(x_1, y_1)[/tex]

Either way, I would like to see the derivation. Any help would be much appreciated.

Thanks!

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# Proving a CDF Identity

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