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  3. 1. a. show that for any y ? rn, show that yyt is p

1. a. Show that for any y ? Rn, show that yyT is positive semidefinite. b. Let X...

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Advance Math

1.

a. Show that for any y ? Rn, show that yyTis positive semidefinite.

b. Let X be a random vector in Rn with covariancematrix ? = E[(X ? E[X])(X ? E[X])T]. Show that ? ispositive semidefinite.

2. Let X and Y be real independent random variables with PDFsgiven by f and g, respectively. Let h be the PDF of the randomvariable Z = X + Y .

a. Derive a general expression for h in terms of f and g

b. If X and Y are both independent and uniformly distributed on[0, 1] (i.e. f(x) = g(x) = 1 for x ? [0, 1] and 0 otherwise) whatis h, the PDF of Z = X + Y ?

Please show your work. Thanks!

Answer & Explanation Solved by verified expert
3.6 Ratings (572 Votes)
1 a A matrix is positivesemidefinite if for all we have Nowlet Then for any we havebecause Hence is positivesemidefiniteb Fix an    See Answer
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