A fixed point of a permutation π:[1, n] →[1, n] is a value for which π(x)=x. Find the variance i

step 1 So if we let x be the number of fixed elements left in the same position of a random permutation

A fixed point of a permutation π:[1, n] →[1, n] is a value...

A fixed point of a permutation $\pi:[1, n] \rightarrow[1, n]$ is a value for which $\pi(x)=x$. Find the variance in the number of fixed points of a permutation chosen uniformly at random from all permutations. (Hint: Let $X_{i}$ be 1 if $\pi(i)=i$, so that $\sum_{i=1}^{n} X_{i}$ is the number of fixed points. You cannot use linearity to find $\operatorname{Var}\left[\sum_{i=1}^{n} X_{i}\right]$, but you can calculate it directly.)

Key Concepts

Covariance

Covariance is a measure of how much two random variables change together. When dealing with sums of indicator random variables, such as in the number of fixed points in a permutation, the covariance between the variables must be considered since the events (or indicators) are not independent. The calculation of overall variance involves summing the individual variances and the covariances between all pairs of these random variables.

Variance

Variance measures the spread or dispersion of a set of random variables around their mean. It quantifies the extent to which the outcomes of a random variable differ from the expected value. In problems involving random permutations, computing the variance of the number of fixed points helps understand how much fluctuation there is around the average count of fixed points.

Permutation

A permutation is an arrangement or reordering of elements from a set, where each element appears exactly once. In the context of mathematics, particularly combinatorics, a permutation of a finite set is a bijective function that reassigns elements to new positions. This concept is fundamental to counting problems and probability theory, as it allows the enumeration of all possible outcomes in a structured way.

Fixed Point

A fixed point in a permutation is an element that remains in its original position after the permutation is applied. Mathematically, for a permutation ? over a set, an element i is a fixed point if ?(i)=i. This property is significant in the analysis of random permutations and has important implications in study of derangements and other combinatorial constructs.

Indicator Random Variable

An indicator random variable is a variable that takes the value 1 if a particular event occurs, and 0 otherwise. In probability and statistics, indicator random variables are used to simplify computations by reducing events to binary outcomes. They are especially useful when summing across events to count occurrences, as in counting the number of fixed points in a permutation.

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