Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2016 • Question 3

Question

Math 6010 • Linear Models Fall 2016 • Question 3

Let \(n\geq 1\) be a fixed integer and recall that there are \(n!\) permutations of \([n]:=\{1,\ldots ,n\}\). Let \(\sigma :=\{\sigma (1),\ldots ,\sigma (n)\}\) denote an arbitrary permutation of \([n]\), and choose and fix some integer \(j\in [n]\). We say that \(\sigma \) fixes \(j\) if \(\sigma (j)=j\). Now suppose that \(\sigma \) is selected at random, all permutations of \([n]\) equally likely.

(a) What is the probability that \(\sigma \) fixed \(j\)?

(b) What is the expectation of the number of integers in \([n]\) that are fixed by \(\sigma \)?

Let \(n\geq1\) be a fixed integer and recall that there are \(n!\)
permutations of \([n]:=\{1,\ldots,n\}\). Let
\(\sigma:=\{\sigma(1),\ldots,\sigma(n)\}\) denote an arbitrary permutation of
\([n]\), and choose and fix some integer \(j\in[n]\). We say that \(\sigma\) fixes
\(j\) if \(\sigma(j)=j\). Now suppose that \(\sigma\) is selected at random, all
permutations of \([n]\) equally likely.

\textbf{(a)} What is the probability that \(\sigma\) fixed \(j\)?

\textbf{(b)} What is the expectation of the number of integers in \([n]\) that
are fixed by \(\sigma\)?

Topic Tags

Classical Probability and Inclusion-Exclusion Conditional Expectation and Conditional Independence Convergence in Probability and Almost Sure Convergence Uniform Integrability and Truncation Methods