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We’ll say that a permutation T = (T(1),.….., T(n)) contains a swap if there exist i, je{1,..,n} so that T(i) = j and (j) =i, where i#j. (a) What is the expected number of swaps in a random permutation? (b) What about the variance? (c) In the same spirit as above, we’ll say that T contains a s-cycle if there exist i1,..., is E {1,...,n} with a(i1) = i2, T(i2) = i3,...,T(i,) = ij. Compute the expectation of the number of s-cycles.

We’ll say that a permutation T = (T(1),.….., T(n)) contains a swap if there exist i, je{1,..,n} so that T(i) = j and (j) =i, where i#j. (a) What is the expected number of swaps in a random permutation? (b) What about the variance? (c) In the same spirit as above, we’ll say that T contains a s-cycle if there exist i1,..., is E {1,...,n} with a(i1) = i2, T(i2) = i3,...,T(i,) = ij. Compute the expectation of the number of s-cycles.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 31E
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We'll say that a permutation T = (T(1),..., T(n)) contains a swap if there exist i, j E {1,.,n} so that T(i) = j and A(j) =i, where i + j. .. . (a) What is the expected number of swaps in a random permutation? (b) What about the variance? (c) In the same spirit as above, we'll say that T contains a s-cycle if there exist i1,..., is E {1,...,n} with T (i1) = i2, T (i2) = i3,..., T(i,) =i1. Compute the expectation of the number of s-cycles. •...
Transcribed Image Text:We'll say that a permutation T = (T(1),..., T(n)) contains a swap if there exist i, j E {1,.,n} so that T(i) = j and A(j) =i, where i + j. .. . (a) What is the expected number of swaps in a random permutation? (b) What about the variance? (c) In the same spirit as above, we'll say that T contains a s-cycle if there exist i1,..., is E {1,...,n} with T (i1) = i2, T (i2) = i3,..., T(i,) =i1. Compute the expectation of the number of s-cycles. •...
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