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3. A matrix S is non-negative—deﬁnite if CTSC Z O for any vector c. Prove that all covari-ance matrices are non-negative—deﬁnite. If CTSC &gt; 0 the matrix is positive deﬁnite.Show that a positive deﬁnite matrix is invertible. Also, give an example of a covariance matrix that is not positive deﬁnite. 4. Let 2 be a random vector of length n that has a M VN (0n,In) distribution.

Let y : AZ + b. Derive y’s moment generating function. What is y’s distribution? 5. Let 2 be a random vector of length n that has a M VN (OMIR) distribution. LetU : ZTZ. Show that Us moment generating function is (1 — 273)_””/2 (for 1 — 215 &gt; 0).This is the MGF for a distribution that has two names. What are they, and what are their parameters?

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