Section 10.2

Indicator Variables

E[sum] = sum of probabilities: expected runs, adjacent pairs, couples seated together, fixed points, intact pairs, peaks and records, crossings of random chords.

Toolkit

Linearity of expectation turns any count of a structural feature into a sum of probabilities: if $X=\sum_i \mathbf 1_{A_i}$ , then $\mathbb{E}[X]=\sum_i \mathbb{P}(A_i)$ , regardless of dependence among the $A_i$ . The method is to choose indicators matching the objects being counted (positions, adjacent pairs, subsets, fixed points), compute each $\mathbb{P}(A_i)$ by a local symmetry or small combinatorial argument, and sum — joint distribution of the $A_i$ is never needed.

Useful recurring facts:

\mathbb{E}\Big[\sum_i \mathbf 1_{A_i}\Big]=\sum_i \mathbb{P}(A_i),\qquad \mathbb{P}(\text{random permutation fixes } i)=\frac1n,

\mathbb{P}(i\text{ is a record among first }i)=\frac1i,\qquad \mathbb{E}[\#\text{runs}]=1+\sum_{i}\mathbb{P}(\text{change at }i).

Exchangeability makes marginal probabilities/means identical across symmetric positions even under strong dependence.

Problems