15. The random vector (X, Y) is said to be uniformly distributed over a region R in the plane if, for some constant

c, its joint density is

$$f(x, y) =

\begin{cases}

c &\text{if } (x, y) \in R \\

0 &\text{otherwise}

\end{cases}

$$

(a) Show that 1/c = area of region R.

Suppose that (X, Y) is uniformly distributed over the square centered at

(0, 0), whose sides are of length 2.

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

(e) What is the probability that (X, Y) lies in the circle of radius 1 centered at the origin? That is, find P(X² +Y² ≤ 1).