5. Suppose X = (X₁, X2, X3) is a normal random vector with mean and covariancematrix Σ, where211(H)1301 0 1Find the joint distribution of Y₁ = X₁ + X2 + X3 and Y₂ = X₁ − X₂.6. Suppose X is a two-dimensional normal random vector with joint densityƒ (11₁, #₂) = k^¹ exp{-}₂ (2² + 2x² − x122–3r₁ – 202 + 4)-4)}.Find E(X) and Cov(X).μ = (2, 1, 2), Σ=

As per our guidelines we are supposed to answer only one question. 5)We have Y1=X1+X2+X3 and…

Answered: 5. Suppose X = T (X₁, X2, X3) is a…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

See similar textbooks

Similar questions

Let JO, J1,..., J4 independent random variables according to the Ber (r;) law, where i = 0, 1,..., 4, respectively. We define the random variables Xi = min {JO + Ji, 1}, for i = 1, 2, 3, (a) Find the law of Xi , for each i = 1, 2, 3, 4. (b) Find the law of (X1, X2, X3, X4).
8) Consider the data (1,1), (3,9), (4,16), If f[1,3] = 4, f[3,4] = z, f[1,3,4] 1. find z %3D
(b) Let X₁, X₂, X3 be uncorrelated random variables, having the same variance ². Consider the linear transformations Y₁ = X₁ + X₂, Y₂ = X₁ + X3 and Y3 = X₂ + X3 . Find the correlations of Yi, Y; for i #j. (5 marks)
3
7. For simple linear regression, we assume that Y = Bo + BIX +e, where e - N(0,0?) and X is fixed (not random). We collect n i.i.d, training sample (x),y1)....(XYn)). Prove that the (Bo.B1) estimated through minimizing RSS equals to the one through maximizing likelihood.
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
4. Let B be an n-dimensional random vector of zero mean and positive- definite covariance matrix Q. Suppose measurements of the form y = WB are made where the rank of W is m. If ß is the linear minimum variance estimate of ß based on y, show that the covariance of the error B - B has rank n - m.
Urgent help is neede please
Let B be an n-dimensional random vector of zero mean and positive- definite covariance matrix Q. Suppose measurements of the form y = WB are made where the rank of W is m. If B is the linear minimum variance estimate of B based on y, show that the covariance of the error B - B has rank n - m.
Example 18.2 For the probability density of a system of random variables (X, Y): f(x, y) = 0.5 sin (x + y) (0 < x < 7,0 << 7), 2' determine (a) the distribution function of the system, (b) the expectations of X and Y, (c) the covariance matrix.
2. (a) When two random variables (X, Y) follow a bivariate normal distribution, the covariance matrix Σ is defined as Σ ρσισ2 (010₂ Posts), - = where p is the correlation, o, o are the variances of X and Y respectively. Use mvrnorm from the MASS library (use library(MASS)) to generate 100 samples from a bivariate normal Call the first column x distribution with μ = (₁, ₂) with μ₁ = (11). 2,42 = 3, and Σ = = and the second column y. (b) Plot the histogram of x and overlay it with the corresponding marginal normal density. Repeat for y. (Recall the marginal distribution of X is N (μ₁, 0²).) (c) Produce a scatter plot of x and y (use plot). Compute the sample correlation coefficient (use cor) and compare with the population correlation p. (First work out p in the Σ given.)
Are X, Y and Z independent?

Recommended textbooks for you

MATLAB: An Introduction with Applications

Statistics

ISBN:

9781119256830

Author:

Amos Gilat

Publisher:

John Wiley & Sons Inc

Probability and Statistics for Engineering and th…

Statistics

ISBN:

9781305251809

Author:

Jay L. Devore

Publisher:

Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…

Statistics

ISBN:

9781305504912

Author:

Frederick J Gravetter, Larry B. Wallnau

Publisher:

Cengage Learning

MATLAB: An Introduction with Applications

Statistics

ISBN:

9781119256830

Author:

Amos Gilat

Publisher:

John Wiley & Sons Inc

Probability and Statistics for Engineering and th…

Statistics

ISBN:

9781305251809

Author:

Jay L. Devore

Publisher:

Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…

Statistics

ISBN:

9781305504912

Author:

Frederick J Gravetter, Larry B. Wallnau

Publisher:

Cengage Learning

Elementary Statistics: Picturing the World (7th E…

Statistics

ISBN:

9780134683416

Author:

Ron Larson, Betsy Farber

Publisher:

PEARSON

The Basic Practice of Statistics

Statistics

ISBN:

9781319042578

Author:

David S. Moore, William I. Notz, Michael A. Fligner

Publisher:

W. H. Freeman

Introduction to the Practice of Statistics

Statistics

ISBN:

9781319013387

Author:

David S. Moore, George P. McCabe, Bruce A. Craig

Publisher:

W. H. Freeman

SEE MORE TEXTBOOKS

5. Suppose X = T (X₁, X2, X3) is a normal random vector with mean and covariance matrix X, where 2 1 30 1 0 1 Find the joint distribution of Y₁ = X₁ + X₂ + X3 and Y₂ = X₁ - X2. 1 1 μ = (2,1,2)T, Σ