Chapter 6, Problem 6.15P

The random vector $\left(X,Y\right)$ is said to be uniformly distributed over a region R in the plane if, for some constant c, its joint density is $f\left(x,y\right)=\{\begin{array}{cc}c& \text{if }\left(x,y\right)\in R\\ 0& \text{otherwise}\end{array}$

a. Show that $\frac{1}{c}=\text{area of regopm R}$. Suppose that $\left(X,Y\right)$ is uniformly distributed over the square centered at (0, 0) and with sides of length 2.

b. Show that X and Y are independent, with each being distributed uniformly over $\left(-1,1\right)$.

c. What is the probability that $\left(X,Y\right)$ lies in the circle of radius 1 centered at the origin? That is, find $P\left\{X^2+Y^2\le 1\right\}$.