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.
Homework Answers
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Let (?,?) have a bivariate normal distribution with mean (0,0) and covariance matrix . Let (?1,?1),…,(??,??)...
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.
Q1. Assume that (XiX2) is multivariate normal with mean vector (0,0) and the variance covariance ...
Q1. Assume that (XiX2) is multivariate normal with mean vector (0,0) and the variance covariance matrix Find the VaRY(p) and ESY(p), where Y = X1 + X2.
Q1. Assume that (XiX2) is multivariate normal with mean vector (0,0) and the variance covariance matrix Find the VaRY(p) and ESY(p), where Y = X1 + X2.
5. Let be a normal random vector with the following mean and covariance matrices: 2 Let also Y; Y...
5. Let be a normal random vector with the following mean and covariance matrices: 2 Let also Y; Y3 where (a) Find P(X2 >0). b Find my EY]. the expected value vector of Y. (c) Find CY, the covariance matrix of Y d) Find P(Y 2).
5. Let be a normal random vector with the following mean and covariance matrices: 2 Let also Y; Y3 where (a) Find P(X2 >0). b Find my EY]. the expected value vector of Y....
3. For n 2 2, let X have n-dimensional normal distribution MN(i, V). For any 1...
3. For n 2 2, let X have n-dimensional normal distribution MN(i, V). For any 1 3 m < n, let X1 denote the vector consisting of the last n - m coordinates of X < n, let 1 (a). Find the mean vector and the variance-covariance matrix of X1. (b). Show that Xi is a (n- m)-dimensional normal random vector.
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...
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...
8. An important distribution in the multivariate setting is the multivariate normal distribution. Let X be...
8. An important distribution in the multivariate setting is the multivariate normal distribution. Let X be a random vector in Rk. That is Xk with X1, X2, ..., xk random variables. If X has a multivariate normal distribution, then its joint pdf is given by f(x) = {27}</2(det 2)1/2 exp {=} (x – u)?g="(x-1)} is the covariant matrix. Note with parameters u, a vector in R", and , a matrix in Rkxk that det is the determinant of matrix ....
4. Let (X,Y) be a bivariate normal random vector with distribution N(u, 2) where -=[ 5...
4. Let (X,Y) be a bivariate normal random vector with distribution N(u, 2) where -=[ 5 ], = [11] Here -1 <p<1. (a) What is P(X > Y)? (b) Is there a constant c such that X and X +cY are independent?
Let Xi,, Xn be a random sample of size n from the normal distribution with mean...
Let Xi,, Xn be a random sample of size n from the normal distribution with mean parameter 0 and variance σ2-3. (a) Justify thatX X, has a normal distribution with mean parameter 0 and variance 3 /n, this is, X~N(0,3/m) (you can do it formally using m.g.f. or use results from normal distribution to justify (b) Find the 0.975 quantile of a standard normal distribution (you can use a table, software or internet to find the quantile). (c) Find the...
1. For each of the following bivariate normal distribution, find its principal components, sketch a typical...
1. For each of the following bivariate normal distribution, find its principal components, sketch a typical scatter plot of data from the distribution, and label the eigenvectors of its covariance matrix on the scatter plot. 9 -5 (a) μ = (3,4),S-1-5 16
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.
Q1. Assume that (XiX2) is multivariate normal with mean vector (0,0) and the variance covariance matrix Find the VaRY(p) and ESY(p), where Y = X1 + X2.
Q1. Assume that (XiX2) is multivariate normal with mean vector (0,0) and the variance covariance matrix Find the VaRY(p) and ESY(p), where Y = X1 + X2.
5. Let be a normal random vector with the following mean and covariance matrices: 2 Let also Y; Y3 where (a) Find P(X2 >0). b Find my EY]. the expected value vector of Y. (c) Find CY, the covariance matrix of Y d) Find P(Y 2).
5. Let be a normal random vector with the following mean and covariance matrices: 2 Let also Y; Y3 where (a) Find P(X2 >0). b Find my EY]. the expected value vector of Y....
3. For n 2 2, let X have n-dimensional normal distribution MN(i, V). For any 1 3 m < n, let X1 denote the vector consisting of the last n - m coordinates of X < n, let 1 (a). Find the mean vector and the variance-covariance matrix of X1. (b). Show that Xi is a (n- m)-dimensional normal random vector.
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...
8. An important distribution in the multivariate setting is the multivariate normal distribution. Let X be a random vector in Rk. That is Xk with X1, X2, ..., xk random variables. If X has a multivariate normal distribution, then its joint pdf is given by f(x) = {27}</2(det 2)1/2 exp {=} (x – u)?g="(x-1)} is the covariant matrix. Note with parameters u, a vector in R", and , a matrix in Rkxk that det is the determinant of matrix ....
4. Let (X,Y) be a bivariate normal random vector with distribution N(u, 2) where -=[ 5 ], = [11] Here -1 <p<1. (a) What is P(X > Y)? (b) Is there a constant c such that X and X +cY are independent?
Let Xi,, Xn be a random sample of size n from the normal distribution with mean parameter 0 and variance σ2-3. (a) Justify thatX X, has a normal distribution with mean parameter 0 and variance 3 /n, this is, X~N(0,3/m) (you can do it formally using m.g.f. or use results from normal distribution to justify (b) Find the 0.975 quantile of a standard normal distribution (you can use a table, software or internet to find the quantile). (c) Find the...
1. For each of the following bivariate normal distribution, find its principal components, sketch a typical scatter plot of data from the distribution, and label the eigenvectors of its covariance matrix on the scatter plot. 9 -5 (a) μ = (3,4),S-1-5 16
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