Exercise 4.11. Again let S be the random walk S, = E- X; with {X;}; i.i.d., takingvalues +1 with p > 1/2 and So = 0. Let u(0) = EeºX1, vo e R. Show that for any 0 E R,the process Mn := e0Sn /u(0)" is a martingale w.r.t. Fn{Ø,N}. Moreover, let 7 =0 > 0, u(0) > 1 and Eu(0)-o(X1, X2, · , Xn), Foinf{n > 0 : Sn = 1}. Note that t < ∞ a.s. Show that for...

This question investigates the martingale property of the random walk Sn and the stopping time…

Answered: Exercise 4.11. Again let S be the…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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5. Let {S, n>0} be a simple random walk with So = 0 and S₁ = X₁ + ... + X, for n ≥ 1, where X₁, i = 1,2,... are independent random variables with P(X; = 1) = p, P(X; = -1) = q = 1 -p for i>1. Assume p‡q. Put Fo= {2,0}, Fn = 0(X₁, X2, ..., Xn), n ≥ 1. Let b, a be two fixed positive integers. Define T = min{n: S₁ = -a or S₁₁=b}. Assume P(T<∞) = 1. Explain why one can use Doob's Optional Stopping Theorem to conclude that E[ZT] = 1. We can use the conclusion that: T is a stopping time with respect to the o-fields Fn, n ≥ 0. A) B) Define Z₁ = (q/p) Sn, n ≥ 0. {Zn, n ≥ 0} is a martingale with respect to the o-fields Fn, n ≥ 0.
5. Let {S, n ≥ 0} be a simple random walk with So = 0 and S₁ = X₁ + ... + X₁, for n ≥ 1, where X₁, i = 1, 2, ... are independent random variables with P(X; = 1) = p, P(X¡ = − 1) = q = 1-p for i > 1. Assume p‡q. Put Fo= {2,0}, Fn = 0(X₁, X2, ..., Xn), n ≥ 1. Let b, a be two fixed positive integers. Define T= min{n: S₁ = -a or Sn=b}. Define Zn = (q/p), n ≥ 0. Prove that {Zn, n ≥ 0} is a martingale with respect to the o-fields Fn, n ≥ 0. We can use the conclusion that: T is a stopping time with respect to the o-fields Fn, n ≥ 0.
Consider a (non)-symmetric random walk M = Xo +X1+X2+…+Xn where X;'s (i> 1) are independent and identical distributed, P(X; = 1) = p and P(X;=-1) =1–p and Xo = 0. Let Sn = eaS,-n Denote Fn= 0(X1,...,Xn). Find a such that (Sn)n20 is a (Fn)n20 - martingale.
Let X1,..., Xn be independent and identically distributed random variables with Var(X1) < ∞. Show that n 1 EjX;→p EX1. п(п + 1) j= Note. A simple way to solve a problem of showing Yn →p a is to establish limn EYn = a and limn Var(Yn) = 0.
4
Exercise 17. Let X₁,..., Xn~ N(u, o2) be i.i.d. where u is known, but o2 is unknown. Consider the estimate n 1 Σ(X₁-1 for the variance. a) Show that 7² C := i=1 -μ)² 72 2 ·Σ( X ₁ = μ)² ~ x² (n). O i=1 b) For a € (0, 1), denote the a-quantile of x2 (n) by qn(a). Using this notation, find numbers an, bn E R such that P(o² bno²) = 2.5%. c) Using these results, develop a test which can be used to test the hypothesis Ho: o² = o against the alternative H₁: 02 08. d) Assume we have n = 100, μ = 0 and we have observed 2 = 4.81. Test the hypothesis Ho: o² = 4 vs. H₁: o² #4 at significance level 5%.
3. Let X Geometric(p) with pe (0, 1). Prove that for every a, beN we have P(X = a+ b|X > a) = P(X = b). 4. Let X be a discrete random variable with the following distribution: P(X = -3) = a, P(X = -2) = ; P(X = 0) = 6: P(X = -1) = b, %3| P(X = 1) = c, P(X = 2) = 32 a. Find a, b, c. b. Calculate E(3+ 2X), E(3+2x)², Var(3+ 2X). 5. Derive the expected value of the random variable X with distribution Negative Binomial(r, p).
Suppose that X1, X2, ..., Xn is a random sample with a probability law P(x) = {0x, 0 Where 8 is an unknown parameter. si x = 1,2,3, 4; otherwise A. Get an estimator of 0 by the method of the moments. B. Prove that the estimator is unbiased.
b) X-XN (0,1), and W₁ σχ YHYN(0,1), for i=1,2,3,...,10, then: Let Z₁ = i) State, with parameter(s), the probability distribution of the statistic, T = ay 10 ΣW2 √Σt,w₁² ii) Find the mean and variance of the statistic T = 10 Σ{12/2 iii) Calculate the probability that a statistic T = Z₁ + W₁ is at most 4. iv) Find the value of ß such that P(T> B) = 0.01, where T = E₁Z²+₁ W².
Let Z₁ = X₁-HX σχ N(0,1), and W₁ = ~N(0,1), for i = 1,2,3,...,10, then: Y₁-Hy OY ΣΤΩ Wee Σ19, 22: ii) Find the mean and variance of the statistic T = 1
Exercise 13. Let X1,..., Xn ~N (μx, σ²) i.i.d. and Y₁,..., Yn ~N(μy, σ²) i.i.d. with known σ². We assume that for each i the random variables X and Y are correlated, with Corr(Xi, Yi) = 1/2. Define Di = Xi - Y₁ for all i = {1,..., n}. Finally, let X, Y and D be the averages of the Xi, Yi and Di, respectively. a) Show that Cov(Xi, Yi) = 0²/2. b) Determine E(D) and Var(Di). c) Consider the test for Ho: px = μy which uses the test statistic Z = √ND and which rejects Ho, if and only if |Z|> 1.96. Show that P(type I error) = 5%. d) In two or three sentences, discuss the differences between the test from part (c) on the one hand, and the non-paired test for comparing the means of two populations on the other
be i.i.d. random variables with expectation 1 and finite 34. Let X1, X, variance o, and set S, = X1+ X2 +-+X,, for n21. Show that VS,-Vn N (0,6*) as and determine the constant 6. and positive, finite

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