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Probability of an event $B$ in case of a discrete pair random variable $$ P[B] = \sum_{(x,y)\in B} P_{X,Y} (x,y). $$

What is it for an event in case of continuous pair random variable?

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If $X,Y$ have joint density $f(x,y)$ then $$\mathbb{P}((X,Y)\in B) =\iint_{\mathbb{R}^2} \mathbb{1}_B(x,y) f(x,y)dx dy,$$ where $\mathbb{1}_B(x,y)$ is the indicator function of the set $B\subset \mathbb{R}^2$, i.e it is equal to $1$ if $(x,y)\in B$ and zero otherwise.

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  • $\begingroup$ You mean simply integrate between the limits of x and y respectively for which it belongs to the event B? Then it becomes similar to the case of Discrete Random Variable we sum for those values for which (x,y) belongs to B? $\endgroup$ Mar 22, 2019 at 7:05
  • $\begingroup$ @AdarshKumar Precisely, it is analogous. Indicator functions are very useful. Both can also be written as $\mathbb{P}(B)=\mathbb{E}(\mathbb{1}_B)$ when it is understood which probability measure is being used. $\endgroup$ Mar 22, 2019 at 18:14

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