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Let f [n]n] be a permutation. A fixed point of f is an element x e [n] such that f(x)-x. Now cons...
(1) Let f : [n] [n] be a permutation. A fixed point of f is an element x e [n] such that f(x) - x...
(1) 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-n -2? (c) What is the probability that X-n-1? (d) What is the expectation of X? (Hint:...
Problem 5. (5 pts) Expectation Trick, Strong Law and the CLT (a) (2 pts) Let X...
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...
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...
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...
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...
A good explanation of how to answer (b) and (c) will be upvoted :) Q3. The...
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...
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...
(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]...
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) ....
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn...
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.)
Let be a permutation of {1,2,……n}.Let -1 be the (n-1)-tuple with one element from missing. Alice...
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...
(1) 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-n -2? (c) What is the probability that X-n-1? (d) What is the expectation of X? (Hint:...
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...
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) ....
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.)
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...
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