= 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.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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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.
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