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

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

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

Suppose that (X, Y) is uniformly distributed over the square centered at (0, 0) and with sides of length 2.

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

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(c) 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}.