Chapter 5.3, Problem 179E

In statistics, the joint density for two independent, normally distributed events with a mean $\mu =0$ and a standard distiibution $\sigma$ is defined by $p\left(x,y\right)=\frac{1}{2\pi {\sigma }^2}e$ Consider (X, Y). the Cartesian coordinates of a ball in the resting position after it was released from a position on the z-axis toward the xv -plane. Assume that the coordinates of the ball are independently normally distributed with a mean p = 0 and a standard deviation of c (in feet). The probability that the ball will stop no more than a feet from the origin is given by $P\left[X^2+Y^2\le a^2\right]={\displaystyle \underset{D}{\iint }p\left(x,y\right)dydx}$. where D is the disk of radius a centered at the origin. Show that $p\left[X^2+Y^2\le a^2\right]=1-e^{-a^2/2{\sigma }^2}$.