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.
We are asked to find the conditional expectation and conditional covariance of a partitioned multivariate normal. Let's solve it carefully, step by step. We have [ X = \begin{pmatrix} X^{(1)} \ X^{(2)} \end{pmatrix} \sim N_4(\mu, \Sigma), ] with [ \mu = \begin{pmatrix} -4 \ 1 \ \hline 4 \ 0 \end{pmatrix}, \quad \Sigma = \begin{pmatrix} 2 & 0 & 1 & 0 \ 0 & 2 & 2 & 0 \ \hline 1 & 2 & 6 & 1 \ 0 & 0 & 1 & 1 \end{pmatrix}. ] Let’s denote the partitions: [ X^{(1)} = \begin{pmatrix} X_1 \ X_2 \end{pmatrix}, \quad X^{(2)} = \begin{pmatrix} X_3 \ X_4 \end{pmatrix}. ] --- ### Step 1: Partition (\Sigma) The covariance matrix can be partitioned as [ \Sigma = \begin{pmatrix} \Sigma_{11} & \Sigma_{12} \ \Sigma_{21} & \Sigma_{22} _____ ____ _______ ___ ____ ____ ____ _______ ___ _____ _______ _____.
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Let , where
and
.
Then find and
.
Let , where
and
. Check the independence
of (i) X2 and X1 (ii) (X2, X4) and (X1, X3) (iii) (X1, X2) and (X3, X4).
If with
and
. Then find the joint distribution
of X1 + 2X2, 2X1 - X2 and X3.
Obtain the square root matrix corresponding to a matrix . Also
verify that .
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.$
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
.