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 ___ _______ ______ ___ _________ ______.
_____ ______ _______ _________ ________ ______ _____ _______ ________ _____ _______ _______.
__________ _____ ______ ___ _____.
_________ ___ ____ ______ __________ ____ ______ _________ ________.
___ ____ _________ _____ __________ __________ ___ ______ ________ _______ _______ __________.
__________ ______ ____ _________ ____ _________ _______ __________ _____ _____ __________ _____.
____ _______ _______ _____ ___ ________ _____ ______.
____ _______ _____ _________ _________ _______ ________ ________ ____.
______ ____ ___ ____ ______ ______ ________.
______ _________ _________ _____ ___ _____.
______ __________ _______ ____ ____ ______ _________ __________ ________ __________ ____.
_______ ______ ________ _________ __________ ___ _________.
_______ _____ ____ _______ ___ _____ __________ ____ ________ ___ _______ ________.
________ _______ _____ __________ __________ _______.
___ ____ ____ ________ _________ ____ _________.
_______ _____ ___ ________ _____ _______ __________ _________ ________ _________ _________.
__________ ______ _______ ______ ________ __________.
_______ _____ _________ __________ ________ ________ ____.
_______ __________ ___ ___ __________ _________ _____ ________ _______ _________ _____ _______.
___ _______ ____ _______ _________ ____ _______.
_____ ________ _______ _________ ________ __________ _____ ________ ____ _______ ______ _________.
___ _______ ________ __________ ________ ________ _____ ____.
___ _______ __________ ________ _______ _______.
_______ ____ ____ ___ _______ _________ ____ __________ _________ ______.
__________ _____ _______ ___ __________ ____ _________ ___ __________ _____ _____.
_________ ____ ________ _________ _________ ____ __________ ____.
_____ _____ _______ _________ ___ ____ _____.
_________ ______ ____ ___ _________ _______ ____ __________ _______ ________.
_____ _______ __________ _____ _______ ______.
_______ _______ ________ _____ ____ _______ _________.
_________ ______ _________ ________ __________ _____ ________ ________.
_____ ________ _________ ________ ____.
_______ _______ _______ _____ __________ ______ ____.
__________ ________ _______ ________ _____ ________ _______ _____ ______ __________ ___ _______.
________ ___ ________ ____ _____.
___ _____ ___ __________ ______ ________ ___ ___.
_________ __________ ___ ________ ________ _________ _______.
_________ _________ _________ __________ _____ ___.
______ _____ __________ _______ _____ _____ _______ _________ ____ ________ ________ ______.
___ ________ _____ ____ ___ _____ ____ _____.
_____ ________ ___ ______ ___ _________ _______ ________ _______ _______ _________ __________.
_____ __________ _______ __________ ___ ______ ______.
____ ________ ___ ________ ________ __________.
______ _______ _______ ___ _____ ______.
______ ____ _____ __________ _________ __________ _________ ______.
_________ ___ ________ ____ _______ _______.
______ ____ _____ __________ ______ _________ ______ ____.
__________ _____ _____ ________ _______.
______ ___ __________ ______ ____ ______ _______ __________ ______ _____ ____.
________ __________ _____ _________ ____ ___ __________ _____.
_____ ______ ____ ____ __________ __________ ____ _________.
__________ ________ __________ _____ __________ ____ ________ _________ ________ ____.
___ _________ ______ _______ _____ _______ ________ __________ _______ ____ ________.
_______ ___ ________ _______ _____ ________ ___ _________ _____.
___ ____ ___ ______ _______ ______ _________ _________ ________ ____ ________.
____ _________ _________ ______ __________ ________ ______ ____ _______.
___ _________ _____ __________ _________ _______ _____ ________ ____ _______ ___ ____.
________ ___ ______ ______ _____ _______ ____ _____.
__________ _______ _________ ____ _______ _______ _________ _______ _______.
_________ _________ ________ __________ ________ __________ __________ _________ ______ _____ ___ _________.
_________ ______ ________ ___ _______ ________ ______ __________ _________.
_______ ____ _________ _______ __________.
___ _________ __________ ________ ______ ____ _________ _________ ___ _____ __________.
_________ _________ ___ ____ _______ ____ ______ ____ ______ _______ ______ ______.
___ _________ ________ _________ ___.
Get Full Answer on WhatsApp
Let , where
and
. Check the independence
of (i) X2 and X1 (ii) (X2, X4) and (X1, X3) (iii) (X1, X2) and (X3, X4).
Let , where
and
.
Then find and
.
Obtain the square root matrix corresponding to a matrix . Also
verify that .
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
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
)
---