If with
and
. Then find the joint distribution
of X1 + 2X2, 2X1 - X2 and X3.
Let ( \mathbf{X} = \begin{pmatrix} X_1 \ X_2 \ X_3 \end{pmatrix} \sim N_3(\boldsymbol{\mu}, \boldsymbol{\Sigma}) ) with [ \boldsymbol{\mu} = \begin{pmatrix} 2 \ 1 \ 2 \end{pmatrix}, \quad \boldsymbol{\Sigma} = \begin{pmatrix} 5 & 3 & 0 \ 3 & 3 & -2 \ 0 & -2 & 5 \end{pmatrix}. ] We are asked to find the joint distribution of [ \mathbf{Y} = \begin{pmatrix} Y_1 \ Y_2 \ Y_3 \end{pmatrix} = \begin{pmatrix} X_1 + 2X_2 \ 2X_1 - X_2 \ X_3 \end{pmatrix}. ] --- Step 1: Express (\mathbf{Y}) as a linear transformation of (\mathbf{X}) [ \mathbf{Y} = A \mathbf{X}, \quad A = \begin{pmatrix} 1 & 2 & 0 \ 2 & -1 & 0 \ 0 & 0 & 1 \end{pmatrix}. ] --- Step 2: Find the mean vector of (\mathbf{Y}) [ \boldsymbol{\mu}_Y = ______ _________ _______ ______ _________ ______ _________ _____ ______ ____ ___ _______.
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Let , where
and
.
Then find and
.
Let two independent samples of size and
from trivariate normal populations have following mean vectors and variance-covariance matrices:
and
and
, respectively.
---$
$
samples are significantly different from each other at 5% level of significance or not.$
Obtain the square root matrix corresponding to a matrix . Also
verify that .
State whether the following statements are True or False. Give reasons in support of your answer: 5×2=10
(a) The eigenvalues of a positive definite matrix are less than zero.
(b) is an orthogonal and idempotent matrix.
(c) A factor model postulates that a random vector is linearly dependent upon a few observable common factors.
(d) A given px1 vector is a plausible value for the mean of a multivariate normal distribution can be tested using Student's t test.
(e) In single linkage method, we find the minimum distance in the distance matrix and merge the minimum distanced clusters
Let ,
and
be partitioned as
,
and
. Then derive the conditional distribution of
.
If with
and
. Then find the joint distribution
of X1 + 2X2, 2X1 - X2 and X3.
Let , where
and
. Check the independence
of (i) X2 and X1 (ii) (X2, X4) and (X1, X3) (iii) (X1, X2) and (X3, X4).