Review Solutions of Probability and Statistics | MATH 30011, Exams of Mathematical Statistics

Download Review Solutions of Probability and Statistics | MATH 30011 and more Exams Mathematical Statistics in PDF only on Docsity! Probability and Statistics Name: Review Solutions Exam 3 Review Notes: Make sure you can read the table D for the exam. As usual, review your class notes and old quizzes. I’ve only included new questions since the last quiz here. 1a) Assume that f (Bt) = B4t and B0 = 0 where Bt is a Brownian Motion. State Itô’s formula for B4t : B4t = R t 0 4B3sdB (s) + 1 2 R t 0 4  3B2sds = 4 R t 0 B3sdB (s) + 6 R t 0 B2sds b) What is the martingale part in part a?R t 0 4B3sdB (s) 2) De…ne Xj as discussed in class. Xj =  1 if !j = H 1 if !j = T ; where H is heads and T is tails. 3) Let M4 = X1 + X2 + X3 be a symmetric random walk as discussed in class. Draw two di¤erent possible paths of M3: 0 1 2 3 0.0 0.5 1.0 x y , which corresponds to ! = HTH 0 1 2 3 -2 -1 0 x y , which corresponds to ! = TTH 4) Assume that W n (t) = 1p n Mnt; where M is a symmetric random walk. Show how and evaluate the variation and mean of W n (t) : Wn(t) =  1p n Mnt = 1p n Mnt = 0; because Xj = 1 1 2
1 1 2
= 0: 2Wn(t) =  2 1p n Mnt = 1 n 2Mnt = 1 n nt = t; because 2Xj = (1 0) 2 1 2 + (1 0)2 1 2 = 1; and 2Mnt =Pnt j=1  2 Xj = Pnt j=1 (1) = nt: 5) What is one way to tell if a function is not di¤erentiable based on the graph? What does this say about Brownian Motion? If there is a kink in the graph, the function is not di¤erentiable. For instance, f (x) = jxj is not di¤erentiable at 0 because, the graph looks like -4 -2 0 2 4 2 4 x y . Since Brownian Motion has in…nitely many kinks, this means that Brownian Motion is nowhere di¤er- entiable. 6)A researcher test the null hypothesis H0 : 1 = 2 vs. Ha : 1 > 2; 1