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 1 (Warm-up problems)

• [Function of random variable (I)] Show that, if [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are random variables, then so are [math]\displaystyle{ X+Y }[/math], [math]\displaystyle{ XY }[/math] and [math]\displaystyle{ \min\{X,Y\} }[/math].
• [Function of random variable (II)] Let [math]X[/math] be a random variable with distribution function [math]\max(0,\min(1,x))[/math]. Let [math]F[/math] be a distribution function which is continuous and strictly increasing. Show that [math]Y=F^{-1}(X)[/math] be a random variable with distribution function [math]F[/math].
• [Independence] Let [math]\displaystyle{ X_r }[/math], [math]\displaystyle{ 1\leq r\leq n }[/math] be independent random variables which are symmetric about [math]\displaystyle{ 0 }[/math]; that is, [math]\displaystyle{ X_r }[/math] and [math]\displaystyle{ -X_r }[/math] have the same distributions. Show that, for all [math]\displaystyle{ x }[/math], [math]\displaystyle{ \mathbf{Pr}[S_n \geq x] = \mathbf{Pr}[S_n \leq -x] }[/math] where [math]\displaystyle{ S_n = \sum_{r=1}^n X_r }[/math]. Is the conclusion true without the assumtion of independence?
• [Dependence] Let [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be discrete random variables with joint mass function [math]\displaystyle{ f(x,y) = \frac{C}{(x+y-1)(x+y)(x+y+1)} }[/math] where [math]\displaystyle{ x,y \in \mathbb{N}_+ }[/math] (In other words, [math]\displaystyle{ x,y = 1,2,3,\cdots }[/math]). Find the marginal mass functions of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ \mathbf{E}[X] }[/math].
• [Expectation] Is it generally true that [math]\mathbf{E}[1/X] = 1/\mathbf{E}[X][/math]? Is it ever true that [math]\mathbf{E}[1/X] = 1/\mathbf{E}[X][/math]?
• [Entropy of discrete random variable] Let [math]X[/math] be a discrete random variable with range of values [math][N] = \{1,2,\ldots,N\}[/math] and probability mass function [math]p[/math]. Define [math]H(X) = -\sum_{n \ge 1} p(n) \log p(n)[/math] with convention [math]0\log 0 = 0[/math]. Prove that [math]H(X) \le \log N[/math] using Jensen's inequality.
• [Law of total expectation] Let [math]X \sim \mathrm{Geom}(p)[/math] for some parameter [math]p \in (0,1)[/math]. Calculate [math]\mathbf{E}[X][/math] using the law of total expectation.

• [Random number of random variables] Let [math]\displaystyle{ \{X_n\}_{n \ge 1} }[/math] be identically distributed random variable and [math]\displaystyle{ N }[/math] be a random variable taking values in the non-negative integers and independent of the [math]\displaystyle{ X_n }[/math] for all [math]\displaystyle{ n \ge 1 }[/math]. Prove that [math]\displaystyle{ \mathbf{E}\left[\sum_{i=1}^N X_i\right] = \mathbf{E}[N] \mathbf{E}[X_1] }[/math].

Problem 4 (Linearity of Expectation, 12 points)

• [Streak] Suppose we flip a fair coin [math]\displaystyle{ n }[/math] times independently to obtain a sequence of flips [math]\displaystyle{ X_1, X_2, \ldots , X_n }[/math]. A streak of flips is a consecutive subsequence of flips that are all the same. For example, if [math]\displaystyle{ X_3 }[/math], [math]\displaystyle{ X_4 }[/math], and [math]\displaystyle{ X_5 }[/math] are all heads, there is a streak of length [math]\displaystyle{ 3 }[/math] starting at the third lip. (If [math]\displaystyle{ X_6 }[/math] is also heads, then there is also a streak of length [math]\displaystyle{ 4 }[/math] starting at the third lip.) Find the expected number of streaks of length [math]\displaystyle{ k }[/math] for some integer [math]\displaystyle{ k \ge 1 }[/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.

• [Expected Mex] Let [math]\displaystyle{ X_1,X_2,\ldots,X_{100} \sim \mathrm{Geo}(1/2) }[/math] be independent random variables. Compute [math]\displaystyle{ \mathbf{E}[\mathrm{mex}(X_1,X_2,\ldots,X_{100})] }[/math], where [math]\displaystyle{ \mathrm{mex}(a_1,a_2,\ldots,a_n) }[/math] is the smallest nonnegative integer that does not appear in [math]\displaystyle{ a_1,a_2,\ldots,a_n }[/math]. Your answer is considered correct if the absolute error does not exceed [math]\displaystyle{ 10^{-6} }[/math]. (Hint: Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required.)

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, 8 points)

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]. Prove that [math]\displaystyle{ \mathbf{E}[|S_n|] = \Theta(\sqrt{n}) }[/math].