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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___ ___ ___ _______ ____.
_______ _________ ____ __________ ______ ____ __________ _______.
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_________ _____ ___ ______ ___ ___ ___ ___.
________ ______ ________ ______ ____ _____ ____ _____ ______ _________ ____.
_____ _________ ______ _______ __________ _____ ________ _________ _________ ________.
________ ______ _____ ____ ___ ___.
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___ ________ __________ _______ ______ ________ _______.
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_____ _____ ___ _________ _________ _____ ____ _____ _______ __________ ____.
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_____ _________ _____ ______ _____ ________ _________ ___ ________.
________ ____ __________ __________ ________ _____ ________ _______ ________ _____ ___ ______.
____ _________ _________ __________ _________ ___ _________ _______ _________ ________ ________.
________ _______ ___ ______ ____ _________ _______ ______ _________ ___ ______ ____.
________ ____ ________ ______ __________ __________.
______ ___ _______ ________ ____.
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___ _______ ___ ____ ________ _____ ________ ___ _______ ____.
___ ______ _______ ____ __________ _________.
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Obtain the square root matrix corresponding to a matrix . Also
verify that .
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 , where
and
.
Then find and
.
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
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).
Let ,
and
be partitioned as
,
and
. Then derive the conditional distribution of
.