(1) Let f : [n] [n] be a permutation. A fixed point of f is an element x e [n] such that f(x) - x...
Let f [n]n] be a permutation. A fixed point of f is an element x e [n] such that f(x)-x. Now consider random permutations of [n] and let X be the random variable which represents the number of fixed points of a given permutation. (a) What is the probability that X 0? (b) What is the probability that X 2? (c) What is the probability that X--1? (d) What is the expectation of X? (Hint: As usual, express X as...
Problem 5. (5 pts) Expectation Trick, Strong Law and the CLT (a) (2 pts) Let X be the number of fixed points in a random permutation of n elements. We know that EX = 1. Compute EX? and EX”. Hint : Write X as a sum of indicator random variables. The number of fixed points in a random permutation is also known as the matching problem. (b) (1pt) For the same X as in part (a), give Markov and Chebyshev...
Problem 5. (5 pts) Expectation Trick, Strong Law and the CLT (a) (2 pts) Let X be the number of fixed points in a random permutation of n elements. We know that EX = 1. Compute EX2 and EX3. Hint : Write X as a sum of indicator random variables. The number of fixed points in a random permutation is also known as the matching problem. (b) (1pt) For the same X as in part (a), give Markov and Chebyshev...
Problem 5. (5 pts) Expectation Trick, Strong Law and the CLT (a) (2 pts) Let X be the number of fixed points in a random permutation of n elements. We know that EX = 1. Compute EX? and EX3. Hint : Write X as a sum of indicator random variables. The number of fixed points in a random permutation is also known as the matching problem. (b) (1pt) For the same X as in part (a), give Markov and Chebyshev...
Question 6: Let n 2 2 be an integer and let ai,a2,...,an be a permutation of the set (1, 2, . . . ,n). Define ao = 0 and an+1 = 0, and consider the sequence do, 1, d2, l3, . . . , Un, Un+1 A position i with 1 i n is called auesome, if ai > ai-1 and ai > ai+1. In words, i is awesome if the value at position i is larger than both its...
Let be a permutation of {1,2,……n}.Let -1 be the (n-1)-tuple with one element from missing. Alice shows Bob -1[i] one by one in the increasing order of i from 1 to (n-1).bob’s task is to compute the missing element from -1 that is in with very limited – O(log n) bits – of memory. Design an algorithm to compute the missing element in this memory-limited and access-limited model, i.e Alice can only show each number to Bob once, and Bob...
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn by, for all a є z /([a]mn) = ([a]rn , [a]n). (a) Prove that f is well-defined. (b) Let m- 4 and n - 7. Find a Z such that f ([al28) (34,(517). (c) Prove that f is a bijection.2 (HINT: To prove that f is onto, given (bm, [cm) E Zm x Zn, consider z - cmr + bns, where 1 mr +ns.)
A good explanation of how to answer (b) and (c) will be upvoted :) Q3. The aim of this question is to see a typical use-case for the linearity of expectation. Consider an experiment in which we toss a biased coin (probability of heads = p) n times. Let Y be the random variable that is the number of heads. Also, let X; be the 0/1 random variable that is 1 if the ith toss was heads and otherwise. b....
Problem 7. (20 pts) Let n N be a natural number and X a finite set with n elements. Show that the number of permutations of X such that no element stays in the same position is n. n! k! For instance, there are 6-3! permutations of 3 elements, but only 2 of them are permutations which fix no element. Similarly, there are 24 41 permutations of 4 elements, but only 9 which fix no element Hint: Use the Inclusion-Erclusion...
Advanced Calculus (3) Let the function f(x) 0 if x Z, but for n e z we have f(n) . Prove that for any interval [a3] the function f is integrable and Ja far-б. Hint: let k be the number of integers in the interval. You can either induct on k or prove integrability directly from the definition or the box-sum criterion. (3) Let the function f(x) 0 if x Z, but for n e z we have f(n) ....