=5. Let {Sn, n ≥ 0} be a simple random walk with So 0 and S₁ = X₁ + ... + X₁, for n ≥ 1, whereX₁, i = 1,2,... are independent random variables with P(X₂ = 1) = p, P(Xį = − 1) = q = 1 -p fori > 1. Assume p‡q. Put Fo= {₁0}, Fn = 0(X₁, X2, ..., Xn), n ≥ 1. Let b, a be two fixed positiveintegers. DefineT = min{n: Sn = -a or Sn=b}.i) Show that I is a stopping time with respect to the o-fields Fn, n ≥ 0.

Given Information: Consider Sn,n≥0 be a simple random walk with S0=0 and Sn=X1+...+Xn for n≥1 Where,…

Answered: 5. Let {Sn, n ≥ 0} be a simple random…

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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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,..., X10 be a random sample of size 10 from a N(u, o²) population. Suppose 10 Y =(X; - 4)? i=1 Find the probability that the random interval Y Y 20.5' 3.25 includes the point o?.
Part II: Sections 2.1 2.7 7. Suppose that X is a discrete random variable with pmf f(x) =D c.14-피 . 뉴- 21 for r=-3, 3, 6, where c is a constant. (a) Find c. Compute E ( ~LX +) (b) 3.
9. If X and Y are two random variables and let g(X) be a random variable. Show that (a) E[g(X) X=x] = g(x). (b) E[g(x)Y|X=x] = g(x) E[Y|X=x]. Assume that E[g(x)] and E[Y] exist.
2. Suppose that X and Y are independent random variables, both with the standard normal distribution i.e. X, Y~ N(0, 1). For pe [-1, 1], define Z:=√1-p²X + pY. Show that E[Z|Y] =pY.
B4. Let X₁,... Xn ~ N(μ, o2) be independent random variables. 2 (a) From lectures we know (X=X) ²³. X₂ (b) Let n i=1 ~ x²(v). What is the value of v? n 1 *³ =, ²-₁, [(x₁ - x) ². Σ(x s n 1 i=1 Determine Var(s), that is, the variance of the sample variance. (c) Assume now that we have observed data ₁,...,n ER with sample variance s². Use the result from part (a) to find numbers an and b, such that [ans, bns] is an exact 95%-confidence interval for o².
Suppose X1,X2.X10 are independent N(H,02) random variables. Z1,Z2.Z5 are independent N(0,1) random variables. The distribution of X + Z is a. N(u, O b. None of these С. N(u,- 10 d. N(0,
I roll a die with 10 faces numbered 0-9 and obtain a score X. What is E(X)? In order to obtain a random number between 0 and 99 I roll the die twice obtaining scores X and Y, then set Z = 10X + Y (i.e., the first roll determines the "tens", the second roll the "ones"). Is it true to say that E(Z) = 10E(X) + E(Y)?
Assume that M= N(h)+k, h² + k, h* + ... Use the following data to find N,(). N,(h)=2.451, N,G)=2.896, N,)=3.285, N,(- = 3.786, N,()= 4.422 16 O a. 4.634 O b. 4.287 O c. 3.044 O d. 3.4146 O e. 3.953 O f. 5.058 Og. 3.674
-) Suppose X and Y are continuous random variables. The range of X is [1,3], the range of Y is [0, 1]. The joint pdf of X and Y be given by f(x, y) = 2xy³ - 2y³. Verify if X and Y independent random variables.
Q 6.1. Suppose Z = (Z1, Z2, Z3) is a standard multi-variate Gaussian random variable i.e., for i ≤ 3, Zi~ N(0, 1) are i.i.d. random variables. Each of the random variables, (a)-(d), on the left is equal in distribution to exactly one random variables, (1)-(4), on the right. Pair up according to "equal in distribution" and explain briefly your reasoning. (a) (X₂) (¹) (X) = (2) 1/√2 (Ⓒ) (x₂) - (3/1² ¹1/1²) (2) (c = 2 0 = 2 Z₁ 1) (²) 3 (4¹) (X) = (²¹) (1) (2) (3) Y₁ 1 (29) - ()) (2) = Y₂ 1 Y₁ (121) = (1/² √₂) (2) (1/√2 1/√2 /2 −1/√√2, Y₂ X₁ X₂ = 22 1 1 2 0 Y₁ 2 1 (4) (2)) = (²) (²) 1 1 Z₁ Z₂ Z3
Researchers wants to determine if low-fat diets or low-carb diets are more effective for weight loss. Suppose 100 people were placed on a low-fat diet and at the end of 3 months the mean weight loss was 10.5 lbs witha standard deviation of 12.5 lbs. There were another 100 people placed on a low-carb diet and at the end of 3 months the mean weight loss was 16.5 lbs with a standard deviation of 10.2 lbs. Does mean weight lossdiffer between the two diets? What are the hypotheses??

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