Question

4. Uniform Stick-Breaking A point X is chosen uniformly from the interval (0, 10) and then a point Y is chosen uniformly from the interval (0, X). This can be imagined as snapping a stick of length 10 and then snapping one of the broken bits. Such processes are called stick-breaking processes. a) Find E(X) and Var(X). See Section 15.3 of the textbook for the variance of the uniform. b) Find E(Y) and Var(Y) by conditioning on X.

Answer 1

6 = 10 (a). we are given that X nunif (0,10), a=0 Hence E(X) = 0410-5 Var (X) = (6-)2 = (0)2 = 8.33 (6) now yn unif Coix) we first find conditional PDF of y given X ł orysa fxlx (7/1) = lor else.

Now E (VW) = fx fox Cytans day - azo, 6=x Var (Yix) = (b-d 12 Now we use law of iterative expectations E(X= E(E(7\x)) = E(9) = { E(X) = {=2.5 and law of iterated variance is Var (y) = E{ var(yıx}4 Var [ Ecv\x)] = E [ ] +Var( at 'Sxtati dok + 1x2.33 *° x10 + .33 120X3 = 6.944