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

Transcribed Image Text:

c if(x, y) e R 0 otherwise