Exercise 13. Let X₁,..., Xn~ N(μ, 0²) i.i.d. and Y₁,..., Yn~ N(µy, o²) i.i.d. with known σ².We assume that for each i the random variables X, and Y₁ are correlated, with Corr(X₁, Y₁) = 1/2.Define D₂ = X₁ - Yį for all i € {1,...,n}. Finally, let X, Y and D be the averages of the X₁, Yiand Di, respectively.a) Show that Cov(X₁, Yi) = 0²/2.b) Determine E(D₁) and Var(D₁).c) Consider the test for Ho: a = μy which uses the test statisticZ = √√nand 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 onehand, and the non-paired test for comparing the means of two populations on the otherhand.2

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Answered: Exercise 13. Let X₁,..., Xn~ N(x, 02)…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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Q7. Let X; (i = 1,.,n) be a random sample from the N(u, o²) population with unknown parameters u and o?. Then we know that T, = E-1(X - X)²/(n – 1) is an unbiased estimator -xi-1 . Now consider two other estimators T, and T3 for o? where of a? and further, (n-1)T T2 = E(X – X)*/n and T3 = E(X – X)²/n + 1). %3D What is the variance of T,? Find out the variances and biases of T2 and T3 and thus obtain their m.s.e.'s. Show that among the three estimators T,, T2 and T3 the one with minimum m.s.e. is T3.
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3 Let X - exp (1). We define Y = [X] +1 and Z = X - [X], where [t] stands for the integer part of t (e.g., [T] = 3). 1. Find the distribution of Y. 2. Compute the expectancy and variance of Y. 3. Compute the CDF of Z.
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%.
Which of the following is false for continuous and independent random variables Y and Z?
Let X; (i = 1, 2,..., n) be i.i.d. exponential random variables with E{X;} = 1/A. Let Y = mn{X1, X2,..., Xn}. What is the distribution of Y? (Compute P{Y > y}.) What is the distribution of Z = max{X1, X2, ..., Xn}? (Compute P{Z < z}.)
3. If X;~N(0,1), i = 1,2,3,4 are independent Normal random variables and Y is given as Y =(X,+X,)' +(X, + X,)' . If aY~x², . find the constant a. 2
We have the following information about the random variables X and Y: 0.5, дY – 1,o = 1, o = 2.25. Calculate the variance of Z 1X + 7Y, | a) when the coefficient of correlation is p(X, Y) = 0.33: b) when X and Y are 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. X₁ (a) X₂ (b) (x₂) - (2) (c) (x²) - (¹/1² 1/√²) (2₁) (3/√2 = √2 = (V2) Z₁ (d) (x) - (1) (²) X2 Y₁ (9)-(-) (2) (1=1) Y₂ (1) Y₁ (²) (4) - (1/² (3) (1/√√2 1/√2-1/√2) (x₁) = (²₁ ² ✓/³²) (3) 2 2 √2 1 Z3 Y₁ → ()-()) (4) = Y₂ 1/2) (2) 1) (2)
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
Suppose that the random variables Y1,..., Y(n > 2) satisfy Y - Ba, + Ei, i= 1,2,..., T, where 1,..., En are iid N(0, o), and both B and o are unknown. (a) Assume z1,..., In are fixed known constants. Here we observe Y1 = n,., Y observed data Y = y = (y1,..., Yn). Find a two-dimensional sufficient statistic of Y = (Yı,., Yn) for (8,0?). (b) Assume now that a1,...,n are random variables with a known joint distribution m(r1,..., In), and the r,'s are independent of G's (it is traditional in the linear regression to use lower case for independent variables r;'s). In this case, the observed data (Y, x) = {(Yi, r;)}i-1,n. Find a three- dimensional sufficient statistic of (Y, x) for (B,02). Im, and the
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