Let ,
and
be partitioned as
,
and
. Then derive the conditional distribution of
.
Let ( X \sim N_p(\mu, \Sigma) ) be partitioned as [ X = \begin{pmatrix} X^{(1)} \ X^{(2)} \end{pmatrix}, \quad \mu = \begin{pmatrix} \mu^{(1)} \ \mu^{(2)} \end{pmatrix}, \quad \Sigma = \begin{pmatrix} \Sigma_{11} & \Sigma_{12} \ \Sigma_{21} & \Sigma_{22} \end{pmatrix}, ] where ( X^{(1)} ) is ( k \times 1 ) and ( X^{(2)} ) is ( (p-k) \times 1 ). We want the conditional distribution _______ _________ _____ ___ ________ _________ _____ ___ _____ ________ _____.
____ _________ _________ _______ __________ _________.
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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.$
Define clustering. Differentiate between single linkage and complete linkage method of clustering.
(b) Consider the following partitioned sample variance-covariance matrix S, obtained from a sample of size 10.$
If , then test for the independence of the sub-vectors
and
.
(Use
)
---
Let ,
and
be partitioned as
,
and
. Then derive the conditional distribution of
.
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
Obtain the square root matrix corresponding to a matrix . Also
verify that .
If with
and
. Then find the joint distribution
of X1 + 2X2, 2X1 - X2 and X3.