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 $\underline{X}\text{and}\underline{Y}$ is symmetric.

(b) $if\underline{x}$ is a p-variate normal random vector, then every linear combination $\underline{c{}'}$ $\underline{x}$ 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) $\text{If}\underline{X}\sim N_2\left(\begin{pmatrix}2\\1\end{pmatrix},\begin{pmatrix}1&0\\0&1\end{pmatrix}\right)\text{and}\underline{Y}\sim N_2\left(\begin{pmatrix}-1\\3\end{pmatrix},\begin{pmatrix}1&0\\0&1\end{pmatrix}\right),\text{then}$

$\underline{X}+\underline{Y}\sim N_2\left(\begin{pmatrix}1\\4\end{pmatrix},\begin{pmatrix}2&0\\0&2\end{pmatrix}\right).$

29 Apr 2025

Answer :

Word Count : 162

(a) False. The matrix $\operatorname{Cov}(\underline X,\underline Y)=E[(\underline X-\mu_X)(\underline Y-\mu_Y)']$ need not equal its transpose; $\operatorname{Cov}(\underline X,\underline Y)'=\operatorname{Cov}(\underline Y,\underline X)$. It is symmetric only when $\underline X=\underline Y$ (or when $\operatorname{Cov}(\underline X,\underline Y)=\operatorname{Cov}(\underline _____ _____ ___ __________ _______ ___ __________ ____ ______ ____ _________ ________.
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x₁	x₂	x₃	x₄	y₁	y₂	y₃	y₄
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	5.56
5	7	5	8	5.68	4.74	5.73	5.56