Let X₁, X₂,..., X20 be a random sample of size n = 20 from an No(μ, Σ) population.Specify each of the following completely.µ)'Σ¯¹(X₁ μ)(a) The distribution of (X₁(b) The distributions of X and √ñ(X – µ)(c) The distribution of (n − 1) S-

Given:1. are random samples from a multivariate normal population with mean vector and covariance…

Answered: Let X₁, X2,..., X20 be a random sample…

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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2. Let X,, X,.. X, be a random sample from a distribution with p.d.f. f(x:0)=0 x, '1>x>0 Is the MLE Ô EE of ? AEE of 0?
7. Let X1, X2, X3 denote a random sample of size 3 from a gamma distribution with a = 7 and 0 = 5 (a). Find the mgf of Y = X1 + X2 + X3. (b). How is Y distributed? (c). How is X distributed?
Let X₁,..., Xn be a random sample from a distribution with one of two pdfs. If 0 = 1, then f(x;0=1) = 1 (0 < x < 1). If 0 = 2, then f(x;0= 2) = 2x1 (0 < x < 1). (a) Give a general form of the MLE of 0. (b) You are given a sample data of 3 values: 0.4, 0.5, 0.8. Find the MLE estimate of 0 based on the data.
(a) Let X1 ~ x² (n, 8) and X2 ~ x² (m, 1). mX1 and its parameters. nX2 (i) Find the distribution of X = (ii) If 1 = 0, name the distribution of X given in 3(a)(i).
Let the following simple random sample X1, X2, · · , X11 following: 1. Binomial pmf. (11, ¾); 2. Uniform pmf; 3. Uniform pdf (0, a); 4. Exponential pdf with (µ). Find the corresponding pmf/pdf of Y1 , Y4, Y7 and F(Y;) where Yi < Y½ < .. < Y6 < ..< Yı1.
16. Let X, X2 X, be a random sample from a normal distribution, X, N(6, 25), and denote by X and S2 the sample mean and sample variance. Use tables from Appendix C to find each of the following: (a) P[3 < X < T. (b) P[L860< 3(X-6)/S]. (c) PIS <319375]
Consider a set of data x1, x2, n n i=1 ..., n taken from a population with mean µ. - Show that (x-μ)² = Σ(x₂ − x)² + n(x − µ)². i=1
1. A discrete random variable X follows the Uniform distribution if X takes values x = 1, 2, .., N, with P(X = x) = 1/N. Compute E(X), E(X²) and the variance Var(X). You may use the following identities: п(п + 1) (1) 2 п(п + 1)(2n + 1) (2) i=1
(a) Let Y be a random variable distributed as X. Determine E(Y) in terms of r. (b) Let {X1, X2, . .. , Xn} be a random sample drawn from a normal distirbution with mean u and 1 variance o?. Denote S E-(X; – X)² as the sample standard deviation. Use the 1 n - result in part (a), or otherwise, to find E(S). (c) Find an unbiased estimator for the population standard deviation o.
4-52. If f(x) =e ,- co
B) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.

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Let X₁, X2,..., X20 be a random sample of size n = 20 from an N(μ, Σ) population. Specify each of the following completely. (a) The distribution of (X₁ − µ)'X¯¹(X₁ − µ) (b) The distributions of X and √n(X – µ) (c) The distribution of (n − 1) S -