Question

Let (x, y ) denote the coordinates of a point chosen at random inside a unit circle whose center is at the origin. their joint probability density function is

{ ¹/π when x²+y² ≤ 1
f(x,y) = { 0 otherwise
Find P{Y > X}.

Answer 1

To find P{Y > X}, we need to determine the area under the joint probability density function f(x, y) where y > x.

To find P{Y > X}, we need to determine the area under the joint probability density function f(x, y) where y > x.

Since the joint probability density function is given as ¹/π when x²+y² ≤ 1 and 0 otherwise, we need to find the bounded region where y > x within the unit circle.

The region where y > x lies in the upper half of the unit circle, below the line y = x.

Thus, the probability P{Y > X} is equal to the area of this region within the unit circle.