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 is symmetric.
(b) is a p-variate normal random vector, then every linear combination
is a scalar vector, is also p-variate normal vector
(c) The trace of matrix
(d) If a matrix is positive definite then its inverse is also positive definite.
(e)
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 $\operatorname{Cov}(\underline X,\underline Y)=\operatorname{Cov}(\underline Y,\underline X)$). (b) False (as stated). If $\underline ___ ________ _______ ____ _______ ___ _______ _________ ________ ________.
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(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 $\operatorname{Cov}(\underline X,\underline Y)=\operatorname{Cov}(\underline Y,\underline X)$). (b) False (as stated). If $\underline ___ ________ _______ ____ _______ ___ _______ _________ ________ ________.
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