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1. For each of the following bivariate normal distribution, find its principal components, sketch a typical...
Let (?,?) have a bivariate normal distribution with mean (0,0) and covariance matrix . Let (?1,?1),…,(??,??)...
Let (?,?) have a bivariate normal distribution with mean (0,0) and covariance matrix . Let (?1,?1),…,(??,??) be a random sample of size n from this distribution. Find a sufficient statistic for p.
Bivariate normal distribution
Problem \(1 \quad\) Bivariate normal distributionAssume that \(\boldsymbol{X}\) is a bivariate normal random variable with$$ \boldsymbol{\mu}=E \boldsymbol{X}=\left(\begin{array}{l} 0 \\ 2 \end{array}\right) \quad \text { and } \quad \Sigma=\operatorname{Cov} \boldsymbol{X}=\left(\begin{array}{ll} 3 & 1 \\ 1 & 3 \end{array}\right) $$Let$$ \boldsymbol{Y}=\left(\begin{array}{l} Y_{1} \\ Y_{2} \end{array}\right)=\left(\begin{array}{lr} 1 / \sqrt{2} & -1 / \sqrt{2} \\ 1 / \sqrt{2} & 1 / \sqrt{2} \end{array}\right) \boldsymbol{X} $$a) Find the mean vector and covariance matrix of \(Y\). What is the distribution of \(Y ?\) Are \(Y_{1}\) and...
Problem 1. (Bivariate Normal Distribution) Let Z1, Z2 be i.i.d. N(0,1) distributed random variables, and p be a constan...
Problem 1. (Bivariate Normal Distribution) Let Z1, Z2 be i.i.d. N(0,1) distributed random variables, and p be a constant between –1 and 1. define X1, X2 as: x3 = + VF5223X = v T14:21 - VF52 23 1) Show that, (X1, X2)T follows bivariate Normal distribution, find out the mean vector and the covariance matrix. 2) Write down the moment generating function, and show that when p= 0, X11X2.
I need this solution Homework: HW 7.5 2 of 5 (5 complete) Score: 0 of 4 pts HV X) 7.5.3 Find the principal components o...
I need this solution
Homework: HW 7.5 2 of 5 (5 complete) Score: 0 of 4 pts HV X) 7.5.3 Find the principal components of the sample covariance matrix below. 71.868 6.6 6.6 41.6 The principal component(s) of the sample covariance matrix, ordered from highest variance to lowest variance, is(are) (Round to two decimal places as needed. Use a comma to separate answers as needed.)
Homework: HW 7.5 2 of 5 (5 complete) Score: 0 of 4 pts HV X)...
Below are four bivariate data sets and the scatter plot for each. (Note that each scatter...
Below are four bivariate data sets and the scatter plot for each. (Note that each scatter plot is displayed on the same scale.) Each data set is made up of sample values drawn from a population. 2.0 3.0 3.0 3.0 mo Figure 1.0 7.5 : 201 9.2 3.0 6.9 4. 05. 5. 0 8 .2 EAN Hool 6.0 46 1 7.8 9. 0 6 .2 20.0145 Figure Answer the following questions. The same response may be the correct answer for...
8. Suppose W and Z have a bivariate normal distribution 1 1 2(1-p2) (-2pzw+u2) fzw (z,...
8. Suppose W and Z have a bivariate normal distribution 1 1 2(1-p2) (-2pzw+u2) fzw (z, w) 27T 1 (i) Find the marginal density of W then compute its MGF Mw and use it to find the mean and the variance of W. [3 (ii Find fzw (z|w) and use it to identify the distribution of Z given W = w aW bwhere a, b E R. [2 (iii) Derive the density of Y (iv) Compute the mean and variance...
Consider the following output from R, where I carry out a principal components anal ysis based...
Consider the following output from R, where I carry out a principal components anal ysis based on the correlation matrix associated with ability.cov. For a description of the data, type ?ability.cov in R round cov2cor (ability.covScov),3) general picture blocks maze reading vocab general .000 0.466 0.552 0.340 0.576 0.514 picture 0.466 1.000 0.572 0.193 0.263 0.239 blocks 0.552 0.572 1.000 0.445 0.354 0.356 0.340 0.193 0.445 1.000 0.184 0.219 reading 0.576 0.263 0.354 0.184 1.000 0.791 vocab > #Eigenvalues 0.514...
1.Assume that x has a normal distribution with the specified mean and standard deviation. Find the...
1.Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.) μ = 101; σ = 16 P(x ≥ 120) = 2.Suppose X ~ N(5, 9). What is the z-score of x = 5? (Enter an exact number as an integer, fraction, or decimal.) z =
1. Let Xi l be a random sample from a normal distribution with mean μ 50 and variance σ2 16. Find...
1. Let Xi l be a random sample from a normal distribution with mean μ 50 and variance σ2 16. Find P (49 < Xs <51) and P (49< X <51) 2. Let Y = X1 + X2 + 15 be the sun! of a random sample of size 15 from the population whose + probability density function is given by 0 otherwise
1. Let Xi l be a random sample from a normal distribution with mean μ 50 and...
Below are four bivariate data sets and the scatter plot for each. (Note that each scatter...
Below are four bivariate data sets and the scatter plot for each. (Note that each scatter plot is displayed on the same scale.) Each data set is made up of sample values drawn from a population. y 1.0 7.4 2.0 9.0 3.0 7.0 11 10- 11 102 9 8+ 7+ 8+ 71 61 5 5 41 4.0 5.4 5.0 7.5 6.05.2 7.0 4.5 8.0 7.1 9.0 5.5 10.0 3.9 V 1.0 8.0 2.0 6.9 3.07.3 4.0 6.1 5.0 7.4 6.0...
Problem 1. (Bivariate Normal Distribution) Let Z1, Z2 be i.i.d. N(0,1) distributed random variables, and p be a constant between –1 and 1. define X1, X2 as: x3 = + VF5223X = v T14:21 - VF52 23 1) Show that, (X1, X2)T follows bivariate Normal distribution, find out the mean vector and the covariance matrix. 2) Write down the moment generating function, and show that when p= 0, X11X2.
I need this solution
Homework: HW 7.5 2 of 5 (5 complete) Score: 0 of 4 pts HV X) 7.5.3 Find the principal components of the sample covariance matrix below. 71.868 6.6 6.6 41.6 The principal component(s) of the sample covariance matrix, ordered from highest variance to lowest variance, is(are) (Round to two decimal places as needed. Use a comma to separate answers as needed.)
Homework: HW 7.5 2 of 5 (5 complete) Score: 0 of 4 pts HV X)...
Below are four bivariate data sets and the scatter plot for each. (Note that each scatter plot is displayed on the same scale.) Each data set is made up of sample values drawn from a population. 2.0 3.0 3.0 3.0 mo Figure 1.0 7.5 : 201 9.2 3.0 6.9 4. 05. 5. 0 8 .2 EAN Hool 6.0 46 1 7.8 9. 0 6 .2 20.0145 Figure Answer the following questions. The same response may be the correct answer for...
8. Suppose W and Z have a bivariate normal distribution 1 1 2(1-p2) (-2pzw+u2) fzw (z, w) 27T 1 (i) Find the marginal density of W then compute its MGF Mw and use it to find the mean and the variance of W. [3 (ii Find fzw (z|w) and use it to identify the distribution of Z given W = w aW bwhere a, b E R. [2 (iii) Derive the density of Y (iv) Compute the mean and variance...
Consider the following output from R, where I carry out a principal components anal ysis based on the correlation matrix associated with ability.cov. For a description of the data, type ?ability.cov in R round cov2cor (ability.covScov),3) general picture blocks maze reading vocab general .000 0.466 0.552 0.340 0.576 0.514 picture 0.466 1.000 0.572 0.193 0.263 0.239 blocks 0.552 0.572 1.000 0.445 0.354 0.356 0.340 0.193 0.445 1.000 0.184 0.219 reading 0.576 0.263 0.354 0.184 1.000 0.791 vocab > #Eigenvalues 0.514...
1. Let Xi l be a random sample from a normal distribution with mean μ 50 and variance σ2 16. Find P (49 < Xs <51) and P (49< X <51) 2. Let Y = X1 + X2 + 15 be the sun! of a random sample of size 15 from the population whose + probability density function is given by 0 otherwise
1. Let Xi l be a random sample from a normal distribution with mean μ 50 and...
Below are four bivariate data sets and the scatter plot for each. (Note that each scatter plot is displayed on the same scale.) Each data set is made up of sample values drawn from a population. y 1.0 7.4 2.0 9.0 3.0 7.0 11 10- 11 102 9 8+ 7+ 8+ 71 61 5 5 41 4.0 5.4 5.0 7.5 6.05.2 7.0 4.5 8.0 7.1 9.0 5.5 10.0 3.9 V 1.0 8.0 2.0 6.9 3.07.3 4.0 6.1 5.0 7.4 6.0...
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