Suppose you are throwing darts numbered i = 1,2,... onto a wall with a target. Assign an (x,y) coordinate system to the wall such that the target is at (0, 0). In this coordinate system, each of your darts (Xi, Yi) follows an independent standard normal distribution in each coordinate, i.e., Xi, Yi N(0, 1) and Xi and Yi are independent. Also all darts are independent amongst themselves.

(a) (5 pts) After throwing n darts, what is the density of the distance from the

origin of the "most successful" dart, i.e., the dart which landed closest to the

origin? Inmathematicalterms,letRi =sqrt(X2 +Y2) for i=1,2,...,n. What is

the probability density function of the minimum, R(1) = min{R1, R2, . . . , Rn}?

(b) (5pts) Find the smallest n such that P(R(1) 1) 0.9, i.e., with at least 90% chance, one of your darts falls within a circle of radius 1 around the origin? You may use the fact that ln(10) = loge(10) 2.3.