PROB5 Let U and V be independent r.v's such that the p.d.f of U is fu(u) = { 2 OSU< 27, otherwise. and the p.d.f'of2 is Seu, v>0, fv (v otherwise. Let X = V2V cos U and Y = 2V sin U. Show that X and Y are independent standard normal variables N(0,1).
Suppose that UCC is open and connected and a E U. Let F:= {f € H(U)| Re(f)> 0, f(a) 1} Show that Fis normal.
For some n > 1, let T E End(Pn) be given by T(p) = p'. Show that T is not diagonalizable.
3. [2+2pt] Let n > 2. Consider a matrix A E Rnxn for which every leading principal submatrix of order less than n is non-singular. (a) Show that A can be factored in the form A = LDU, where Le Rnxn is unit lower triangular, D e Rnxn is diagonal and U E Rnxn is unit upper triangular. (b) If the factorization A = LU is known, where L is unit lower triangular and U is upper triangular, show how...
4. Let A = {x EZ|x>10) and B = {x EZ|X<-10). Is A U B countable? Why? 5. What is the term al of the sequence {a") if a" = 2-!?
5. Let S = {vi, u2, , v) be a set of k vectors in Rn with k > n. Show that S cannot be a basis for Rn.
2. Let YBinomial (n, p) where p docs not depend on . Without using the WLLN (but you can use e.g. Chebychev's inequality) show that (a) Y/n-, p as n → oo (b) (1-Y/n)-> (1-p) as n → oo
Let ne N. Show that Zn forn > 2 is not a group under multiplication as defined above. What happens for n = 1?
(5) Use induction to show that Ig(n) <n for all n > 1.
5. Prove that U(2") (n > 3) is not cyclic.