Exercise 14. Let bi,..., bn be distinct numbers, representing the quality of n people. Sup pose n people arrive to interview for a job, one at a time, in a random order. That is, every possible arrival order of these people is equally likely. We can think of an arrival ordering of the people as an ordered list of the form ai, . . . , an, where the list ai, . . . , an is a permutation of...
Problem 3: Let (w).>o be a sequence such that bn is convergent. Let (an)nzo to be a sequence such that lant - and by for all ne N. Prove that (an)nzo is a convergent sequence. (Hint: We did something similar in class before.)
Let ao 2 bo > 0, and consider the sequences an and bn defined by an + bn n20 (1) Compute an+l-bn+1 1n terms of Van-v/bn. (2) Prove that the sequence an is nonincreasing, that the sequence bn Is nonde- creasing, and that an 2 bn for all n 20 (3) Prove that VanVbn S Cr for all n20, where C> 0 and y>1 (give values of C and γ for which this inequality holds). Conclude that an-bn C,γ-n, where...
2. Let {An}n>1 and {Bn}n>ı be two sequences of measurable sets in the measurable space (12,F). Set Cn = An ñ Bn, Dn = An U Bn: (1) Show that (Tim An) ^ ( lim Bm) – lim Cn (lim An) ( lim Bu) C lim Dm and 100 noo (2) Show by example the two inclusions in (1) can be strict.
Let Mi be the set of all sequences {a.);, of real num bers such that Σ converges. More formally, we could write this as 1 lal M1a :(W) ai R and i=1 We introduce a function p: Mi x MiR by setting 95 Let (Mi,p) denote the particular metric space we introduced above, and for each X = {xīた1 e M and for each i, we refer to the number xi as the ith coordinate of X. For each N...
QUESTION 5 Let V denote an arbitrary finite-dimensional vector space with dimension n E N Let B = {bi, bn} and B' = { bị, b, } denote two bases for V and let PB-B, be the transition matrix from B to B' Prove that where 1 V → V is the identity transformation, i e 1(v) v for all v E V Note that I s a linear transformation 14] QUESTION 5 Let V denote an arbitrary finite-dimensional vector...
11. Let an >0 and assume that bn = n+1 + B. What can we say about the convergence of an? an
ame: . (10 points) Let p > 3 be any prime number. (a) Show that p mod 6 is equal to 1 or 5 (b) Use part (a) to prove that pe - 1 is always a multiple of 24.
5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are the quadratic residues of p that lie between 1 and p - 1. Prove that 1,0 (P-1)/2 i- 1 Hint: If a is a quadratic residue less than or equal to (p-1)/2 then what is p - ai? 5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are...
Literally in a desperate situation. Appreciate so much if any expert could give me a hand! Many thanks in advance!!! Will give a thumb up afterwards definitely! Problem 4. Let Xk be an independent identically distributed sequence o continuous real valued random variables on a probability space (Ý, F, P). Suppose that Xk models your result in match k N. We say that you achieve a personal best in m atch n N if Xn > Xk for all 1-k...