Exercise 13. Let X₁,..., Xn~ N(x, 0²) i.i.d. and Y₁,..., Yn ~ N(µy, 02) i.i.d. with known o².We assume that for each i the random variables X; and Y; are correlated, with Corr(Xį, Y₁) = 1/2.Define D₁ = X₁ - Yį for all i = {1,...,n}. Finally, let X, Y and D be the averages of the Xi, Yiand Di, respectively.a) Show that Cov(X₁, Yi) = 0²/2.b) Determine E(D₁) and Var(D₁).c) Consider the test for Ho: x = µy which uses the test statisticZ = √√nDσand 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.

The provided information is as follows: and are i.i.d variables.

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

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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Let X, X2.**An be an i.i.d random sample from U(0,0), then the sufficient statistics for 0 is Select one: O a. h= max(x,) O b. h = min(x) C. no sufficient statistics for 0 d.
Let X1, X2, ... ,Xn form a random sample from U(0,0). Find minimal sufficient statistics.
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Prove that the events Es, E2... En are independent iff their corresponding indicator variates I,, I₂.. In are independent.
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).
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:
Let Y1,..., Y, be i.i.d. random variables from the Weibull distribution f(y; 0) Σ (멤) exp ), y > 0. Show that ô = Li=1í is an unbiased estimator of 0. Is Ô an efficient estimaotr of 0?
prove
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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