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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Obtain the square root matrix corresponding to a matrix . Also
verify that .
Let , where
and
. Check the independence
of (i) X2 and X1 (ii) (X2, X4) and (X1, X3) (iii) (X1, X2) and (X3, X4).
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
)
---
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 , where
and
.
Then find and
.
Let ,
and
be partitioned as
,
and
. Then derive the conditional distribution of
.