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= If f is a permutation of the set X {1, 2,...,n} then a cycle in f is a sequence of distinct elements a₁, a2,..., ak of X with the property that f(a) = a++1 for i = 1,...,k 1 and f(ak) = a₁. The number of elements in the sequence is the length of the cycle. Note that every element of X is in a unique cycle of f and that an element x = X such that f(x) = x is a cycle of length 1. Let be the set of permutations of the set {1, 2,..., n}. Consider the probability space on given by the uniform distribution. Determine the expected length of the cycle containing 1 in a permutation chosen at random from this probability space.

= If f is a permutation of the set X {1, 2,...,n} then a cycle in f is a sequence of distinct elements a₁, a2,..., ak of X with the property that f(a) = a++1 for i = 1,...,k 1 and f(ak) = a₁. The number of elements in the sequence is the length of the cycle. Note that every element of X is in a unique cycle of f and that an element x = X such that f(x) = x is a cycle of length 1. Let be the set of permutations of the set {1, 2,..., n}. Consider the probability space on given by the uniform distribution. Determine the expected length of the cycle containing 1 in a permutation chosen at random from this probability space.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 74E
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= If f is a permutation of the set X {1, 2,...,n} then a cycle in f is a sequence of distinct elements a₁, a2,..., ak of X with the property that f(a) = a++1 for i = 1,...,k 1 and f(ak) = a₁. The number of elements in the sequence is the length of the cycle. Note that every element of X is in a unique cycle of f and that an element x = X such that f(x) = x is a cycle of length 1. Let be the set of permutations of the set {1, 2,..., n}. Consider the probability space on given by the uniform distribution. Determine the expected length of the cycle containing 1 in a permutation chosen at random from this probability space.
Transcribed Image Text:= If f is a permutation of the set X {1, 2,...,n} then a cycle in f is a sequence of distinct elements a₁, a2,..., ak of X with the property that f(a) = a++1 for i = 1,...,k 1 and f(ak) = a₁. The number of elements in the sequence is the length of the cycle. Note that every element of X is in a unique cycle of f and that an element x = X such that f(x) = x is a cycle of length 1. Let be the set of permutations of the set {1, 2,..., n}. Consider the probability space on given by the uniform distribution. Determine the expected length of the cycle containing 1 in a permutation chosen at random from this probability space.
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