Events about pairs of continuous random variables, such as \("X+Y \gt 1"\) or \("X \gt 5 \textbf{ and } Y \lt 3"\) correspond to regions of the x-y plane.
The probability for any such event is the volume under the joint probability density function above this region, and can be evaluated as a double-integral.
Probabilities as integrals
The probability of any event \(A\) about the variables \(X\) and \(Y\) can be evaluated as
\[ P(A) \;\;=\;\; \iint\limits_{(x,y) \in A} f(x,y)\;dx\;dy \]Care must be taken with the integration limits for the inner and outer integrals when evaluating this integral — for some events, the limits for the inner integral must involve the outer integral's variable.
Example
The random variables \(X\) and \(Y\) have joint probability density function
\[ f(x,y) \;=\; \begin{cases} x+y & \quad\text{if }0 \lt x \lt 1 \text{ and }0 \lt y \lt 1 \\ 0 & \quad\text{otherwise} \end{cases} \]What is the probability that \(X+Y\) will be less than one?
(Solved in full version)