Chapter 5.2, Problem 118E

Suppose that (X. Y) is the outcome of an experiment that must occur in a particular region S in the xy —plane. In this context, the region S is called the

sample space of the experiment and X and Y are random variables. If D is a region included in S. then the

probability of (X. Y) being in D is defined as $P\left[\left(X,Y\right)\in D\right]={\displaystyle \underset{D}{\iint }p\left(x,y\right)dxdy}$. where p(x. y) is the joint probability density of the experiment. Here, $p\left(x,y\right)$

is a nonnegative function for which ${\displaystyle \underset{D}{\iint }p\left(x,y\right)dxdy=1}$ .

Assume that a point (X, Y) is chosen arbitrarily in the square $[0,3]\times [0,3]$ with the probability density

$p\left(x,y\right)=\left\{{}_{0\text{otherwise}\text{.}}^{\frac{1}{9}\left(x,y\right)\in \left[0,3\right]\times \left[0,3\right]}\right\}$ , Find the

probability that the point (X. Y) is inside the unit square and interpret the result.