A quarterback throws a football at a target marked out on the ground 40 yards from his position. Assume that the PDF for the football's hitting the target is Gaussian within the plane of the target. Let the coordinates of the plane of the target be denoted by the and axes.
Thus, the joint PDF of is a two-dimensional Gaussian PDF. The average location of the hits is at the origin of the target, and the standard deviation in each direction is the same and is denoted as σ.
Assuming and are independent, find the probability that the hits will be located within an annular ring of width located a distance from the origin; that is, find the probability density function for hits as a function of the radius from the origin.