I'd like to know how to prove (or show that it is reasonable) that the probability that a random vector [itex](X, Y)[/itex] assumes a value in the region [itex]B\subseteq \mathbb{R}^2[/itex] is(adsbygoogle = window.adsbygoogle || []).push({});

[itex](1)[/itex] [itex]Pr((X, Y) \in B)=\iint\limits_B \, f_{X,Y}(x, y) \mathrm{d}x\,\mathrm{d}y[/itex].

My textbook doesn't provide much of an explanation for the above formula except that it is "the volume under the surface defined by the density and lying above the region [itex]B[/itex]." However, the univariate case is explained in the text by appealing to the fundamental theorem of calculus:

[itex](2)[/itex] [itex]Pr(X \in (a, b])=Pr(X \leq b)-Pr(X \leq a) = F_{X}(b)-F_{X}(a) = \int_a^b \! f_{X}(x) \, \mathrm{d}x[/itex].

Since my text practically proves the formula for [itex](2)[/itex] by appealing to the fundamental theorem of calculus, I've been assuming that [itex](1)[/itex] is not a definition and can be similarly proved by some theorem or set of theorems. Is this true, and if so, how can I prove [itex](1)[/itex]?

Also, I should mention that I'm currently taking an introductory course in probability and have no knowledge of measure theory.

Thanks,

Bijan

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# Proof of the formula for the probability in a region

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