5. What conditions on n, k need to hold so that n(n − 1)· · ·(n − k + 1) = n! /(n−k)! is a true statement? Write this as a theorem and prove it
4. Here is a fact about permutations: (*) nPr= n!/(n-k)!, for all k =n. Let's prove this via mathematical induction for the fixed case k-3. 2 of 3 (i) Write clearly the statement (**) we wish to prove. Be sure your statement includes the phrase "for all n" (ii) State explicitly the assumption in (*) we will thus automatically make about k-2 (ii) Now recall that to prove by induction means to show that IfmPm!/lm-k)! is true for all km...
,X, ,n. independent, the central Xi, E(X)=0, var(X)-σ are Prove 3. Assume <o。 13<oo, 1=1, limit theorem (CLT) based EX1 result regarding what are conditions on σ that we need to assume in order for the x.B.= Σσ, as n →oo. In this context, X,, B" =y as n →oo, In this context, result to hold?
need help with a and b in this graph theory question Let n >k> 1 with n even and k odd. Make a k-regular graph G by putting n vertices in a circle and connecting each vertex to the exact a) Show that for all u,v there are k internally disjoint u, v-paths (you (b) Use the previous part, even if you did not prove it, to show that the e vertex and the k 1 closest vertices on either...
Therom 1.8.2 n choose k = (n choose n-k) n choose k = (n-1 choose K) + (n-1 choose K-1) 2n = summation of (n choose i ) please use the induction method (a) (10 pts) Show that the following equality holds: n +1 + 2 Hint: If you proceed by induction, you might want to use Theorem 1.8.2. If you search for a combinatorial proof, consider the set X - (i,j, k): 0 S i,j< k< n) (b) (10...
4) Find the function, f (k,p), such that n” Vp, nc N k=1 Bonus (up to 3 points): 5) Prove the formula in (4). (Hint: maybe try the Binomial theorem)
Recall that Etan E R is positive if the following two conditions hold: There exists N E Z+ such that an >0 for alln2 N. We use the notation R+to denote the set of positive real numbers: R+ = { E{a») R : Efe») is positive} 1. In class, we proved that the relation<on R, given by is an order relation. In this problem, you'll prove that R satisfies the axioms of an ordered field (a) If E(anh E{놔,Ep., }...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
Exercise 5 Consider the multiple regression model y_Xß+e. The Gauss-Markov conditions hold and also ? ~ MVN(0, ?21). Let Q-(C3-7), [C(xx)-cj-? (C3-7) tha EQ m?2 + (something > 0). , where C is an m x (k + 1) matrix. Show Exercise 6 Consider the multiple regression model y_Xß+?. The Gauss-Markov conditions hold and also ? MVN(0, ?21). Evaluate E (YAY-?2)2 where A n-1 RT [I-X(XX)-1X'] .
1. [5 marks Show the following hold using the definition of Big Oh: a) 2 mark 1729 is O(1) b) 3 marks 2n2-4n -3 is O(n2) 2. [3 marks] Using the definition of Big-Oh, prove that 2n2(n 1) is not O(n2) 3. 6 marks Let f(n),g(n), h(n) be complexity functions. Using the definition of Big-Oh, prove the following two claims a) 3 marks Let k be a positive real constant and f(n) is O(g(n)), then k f(n) is O(g(n)) b)...
(1) Using the identity: n n! (2) want k k!(n - k)! for n > 1, prove the following identity: ()-20) + n2