概率论与数理统计 (Spring 2024)/Problem Set 2

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Assumption throughout Problem Set 2

Without further notice, we are working on probability space [math]\displaystyle{ (\Omega,\mathcal{F},\mathbf{Pr}) }[/math].

Without further notice, we assume that the expectation of random variables are well-defined.

The term [math]\displaystyle{ \log }[/math] used in this context refers to the natural logarithm.

Problem 4 (Linearity of Expectation)

• [Inversion] Given a sequence of [math]\displaystyle{ n }[/math] elements [math]\displaystyle{ a_1, a_2, \ldots, a_n }[/math], an inversion is a pair of integer [math]\displaystyle{ (i, j) }[/math] such that [math]\displaystyle{ 1 \le i \lt j \le n }[/math] and [math]\displaystyle{ a_i \gt a_j }[/math]. For instance, in the sequence [math]\displaystyle{ a = [1,2,5,1,3] }[/math] there are three inversions: [math]\displaystyle{ (2,4), (3,4), (3,5) }[/math]. Suppose we choose a sequence [math]\displaystyle{ \rho }[/math] from [math]\displaystyle{ [q]^n }[/math] uniformly at random, where [math]\displaystyle{ n }[/math] and [math]\displaystyle{ q }[/math] are given integers with [math]\displaystyle{ q \ge 1 }[/math]. Find the expected number of inversions in [math]\displaystyle{ \rho }[/math].
• [Number of cycles] At a banquet, there are [math]\displaystyle{ n }[/math] people who shake hands according to the following process: In each round, two idle hands are randomly selected and shaken (these two hands are no longer idle). After [math]\displaystyle{ n }[/math] rounds, there will be no idle hands left, and the [math]\displaystyle{ n }[/math] people will form several cycles. For example, when [math]\displaystyle{ n=3 }[/math], the following situation may occur: the left and right hands of the first person are held together, the left hand of the second person and the right hand of the third person are held together, and the right hand of the second person and the left hand of the third person are held together. In this case, three people form two cycles. How many cycles are expected to be formed after [math]\displaystyle{ n }[/math] rounds?
• [Paper Cutting] We have a rectangular piece of paper divided into [math]\displaystyle{ H \times W }[/math] squares, where two of those squares are painted black and the rest are painted white. If we let [math]\displaystyle{ (i,j) }[/math] denote the square at the [math]\displaystyle{ i }[/math]-th row and [math]\displaystyle{ j }[/math]-th column, the squares painted black are [math]\displaystyle{ (h_1,w_1) }[/math] and [math]\displaystyle{ (h_2,w_2) }[/math]. Bob will repeat the following operation to cut the piece of paper:
  Assume that we have [math]\displaystyle{ h \times w }[/math] squares remaining. There are [math]\displaystyle{ (h−1) }[/math] horizontal lines and [math]\displaystyle{ (w−1) }[/math] vertical lines that are parallel to the edges of the piece and pass the borders of the squares. He chooses one of these lines uniformly at random and cuts the piece into two along that line. Then, if the two black squares are on the same piece, he throws away the other piece and continues the process; otherwise, he ends the process.

  Find the expected value of the number of times Bob cuts a piece of paper until he ends the process.

Problem 5 (Probability meets graph theory)

In this part we will work on undirected simple graphs. For any [math]\displaystyle{ G = (V,E) }[/math], a clique (团，完全子图) is a subset of vertices [math]\displaystyle{ C\subset V }[/math], such that every two distinct vertices of [math]\displaystyle{ C }[/math] are adjacent, and an independent set (独立集) is a subset of vertices [math]\displaystyle{ I\subset V }[/math] such that no two distinct vertices of [math]\displaystyle{ I }[/math] are adjacent.

• [Erdős–Rényi random graph] Consider [math]\displaystyle{ G\sim G(n,p) }[/math] where [math]\displaystyle{ G(n,p) }[/math] is the Erdős–Rényi random graph model.
  1. Let [math]\displaystyle{ p\in (0,1) }[/math]. A "triangle" in a graph is a clique of size [math]\displaystyle{ 3 }[/math]. Find the expected number of triangles in [math]\displaystyle{ G }[/math]. (Hint: use indicators and the linearity of expectation.)
  2. Let [math]\displaystyle{ p\in (0,1) }[/math]. For any [math]\displaystyle{ 2\leq q\leq n }[/math], let the random variable [math]\displaystyle{ N_q }[/math] be the number of [math]\displaystyle{ q }[/math]-cliques. Here, a [math]\displaystyle{ q }[/math]-clique is a clique of size [math]\displaystyle{ q }[/math]. Find [math]\displaystyle{ \mathbf{E}[N_q] }[/math].
  3. Let [math]\displaystyle{ p = 1/2 }[/math]. For an undirected graph [math]\displaystyle{ G }[/math], define [math]\displaystyle{ \alpha(G) = \max\{|S|:S \text{ is an independent set}\} }[/math]. Show that when [math]\displaystyle{ n\rightarrow \infty }[/math], [math]\displaystyle{ \mathbf{Pr}[\alpha(G) \geq 3\log_2 n +1] \rightarrow 0 }[/math]. Also, please interpret this result in the context of social networks, in which the vertices represent people, and the edges represent friendship.
• [Random social networks] Let [math]\displaystyle{ G = (V, E) }[/math] be a fixed undirected graph without isolating vertex. Let [math]\displaystyle{ d_v }[/math] be the degree of vertex [math]\displaystyle{ v }[/math]. Let [math]\displaystyle{ Y }[/math] be a uniformly chosen vertex, and [math]\displaystyle{ Z }[/math] a uniformly chosen neighbor of [math]\displaystyle{ Y }[/math].
  1. Show that [math]\displaystyle{ \mathbf{E}[d_Z] \geq \mathbf{E}[d_Y] }[/math].
  2. Interpret this inequality in the context of social networks, in which the vertices represent people, and the edges represent friendship.
• [Turán's Theorem] Let [math]\displaystyle{ G=(V,E) }[/math] be a fixed undirected graph, and write [math]\displaystyle{ d_v }[/math] for the degree of the vertex [math]\displaystyle{ v }[/math]. Use probablistic method to prove that [math]\displaystyle{ \alpha(G) \ge \sum_{v \in V} \frac{1}{d_v + 1} }[/math]. (Hint: Consider the following random procedure for generating an independent set [math]\displaystyle{ I }[/math] from a graph with vertex set [math]\displaystyle{ V }[/math]: First, generate a random permutation of the vertices, denoted as [math]\displaystyle{ v_1,v_2,\ldots,v_n }[/math]. Then, construct the independent set [math]\displaystyle{ I }[/math] as follows: For each vertex [math]\displaystyle{ v_i \in V }[/math], add [math]\displaystyle{ v_i }[/math] to [math]\displaystyle{ I }[/math] if and only if none of its predecessors in the permutation, i.e., [math]\displaystyle{ v_1,\ldots,v_{i-1} }[/math], are neighbors of [math]\displaystyle{ v_i }[/math].)

Problem 6 (1D random walk)

Let [math]\displaystyle{ p \in (0,1) }[/math] be a constant, and [math]\displaystyle{ \{X_n\}_{n \ge 1} }[/math] be independent Bernoulli trials with successful probability [math]\displaystyle{ p }[/math]. Define [math]\displaystyle{ S_n = 2\sum_{i=1}^n X_i - n }[/math] and [math]\displaystyle{ S_0 = 0 }[/math].

• [Range of random walk] The range [math]\displaystyle{ R_n }[/math] of [math]\displaystyle{ S_0, S_1, \ldots, S_n }[/math] is defined as the number of distinct values taken by the sequence. Show that [math]\displaystyle{ \mathbf{Pr}\left(R_n = R_{n-1}+1\right) = \mathbf{Pr}\left(\forall 1 \le i \le n, S_i \neq 0\right) }[/math] as [math]\displaystyle{ n \to \infty }[/math], and deduce that [math]\displaystyle{ n^{-1} \mathbf{E}[R_n]\to \mathbf{Pr}(\forall i \ge 1, S_i \neq 0) }[/math]. Hence show that [math]\displaystyle{ n^{-1} \mathbf{E}[R_n] \to |2p-1| }[/math] as [math]\displaystyle{ n \to \infty }[/math].
• [Symmetric 1D random walk (III)] Suppose [math]\displaystyle{ p = \frac{1}{2} }[/math]. Let [math]\displaystyle{ T=\min\{n \ge 1:S_n=0\} }[/math] be the time of the first return of the walk to its starting point. Show that [math]\displaystyle{ \mathbf{E}[T^\alpha] \lt \infty }[/math] if and only if [math]\displaystyle{ \alpha \lt \frac{1}{2} }[/math].
• [Symmetric 1D random walk (IV)] Suppose [math]\displaystyle{ p = \frac{1}{2} }[/math]. Prove that [math]\displaystyle{ \mathbf{E}[|S_n|] = \Theta(\sqrt{n}) }[/math].