Suppose we have a stick of length 1 which we break at a point distance

X Unif[0,1] from its left endpoint, creating two pieces.

(a) (4 pts) Let L = max(X, 1 X) denote the length of the longer piece. Show that L Unif[1/2,1]. Hint: Find the CDF, FL(l) = P(max(X,1 X) l), for 21 < l < 1.

(b) (2pts) Suppose we break the longer segment of length L in half and form a triangle out of the 3 resulting pieces. Let be the angle shown in the figure. Show that = arccos(1/L 1) for 0 < < 2 .

Note: the vertical line bisects the bottom side and forms a right angle.

(c) (4pts) Find the density of .