State whether the following statements are true or false and also give

Question

State whether the following statements are true or false and also give the reason in support of your answer

(a) The covariance matrix of random vectors $\underset{\sim }{X}$ and $\underset{\sim }{Y}$ is symmetric.

(b) If $\underset{\sim }{X}$ is a p-variate normal random vector, then every linear combination $\underset{\sim }{C{}'}\underset{\sim }{X},$ where $\underset{\sim }{C}_{P\times 1}$ is a scalar vector, is also p-variate normal vector.

(d) If a matrix is positive definite then its inverse is also positive definite.

(e) If $\underset{\sim }{X}$ $-N_2$ $\left ( \binom{2}{1},\begin{pmatrix} 1 &0 \\ 0 &1 \end{pmatrix} \right )$ and

$\underset{\sim }{Y}$ $-N_2$ $\left ( \binom{-1}{3},\begin{pmatrix} 1 &0 \\ 0 &1 \end{pmatrix} \right ),$ then

$\underset{\sim }{X}$ $+$ $\underset{\sim }{Y}$ $-N_2$ $\left ( \binom{1}{1},\begin{pmatrix} 1 &0 \\ 0 &1 \end{pmatrix} \right )$

02 Apr 2024

Answer :

Word Count : 279

Let's analyze each statement:

(a) True. The covariance matrix of random vectors $\underset{\sim }{X}$ and $\underset{\sim }{Y}$ is given by $Cov(\underset{\sim }{X}, \underset{\sim }{Y}) = E[(\underset{\sim }{X} - \mu_X)(\underset{\sim }{Y} - \mu_Y)^T]$, where $\mu_X$ and $\mu_Y$ are the means of $\underset{\sim }{X}$ and $\underset{\sim }{Y}$ respectively. ________ ___ _________ ___ ________ _________ _______ __________ ____.
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$x_1$	$x_2$	$x_3$	$x_4$	$y_1$	$y_2$	$y_3$	$y_4$
10	10	10	8	8.04	9.14	7.46	6.58
8	8	8	8	6.95	8.14	6.77	5.76
13	13	13	8	7.58	8.74	12.74	7.71
9	9	9	8	8.81	8.77	7.11	8.84
11	11	11	8	8.33	9.26	7.81	8.47
14	14	14	8	9.96	8.10	8.84	7.04
6	6	6	8	7.24	6.13	6.08	5.25
4	4	4	19	4.26	3.10	5.39	12.50
12	12	12	8	10.84	9.13	8.15	5.56
7	7	7	8	4.82	7.26	6.42	7.91
5	5	5	8	5.68	4.74	5.73