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!
