概率论与数理统计 (Spring 2024)/Problem Set 1

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Assumption throughout Problem Set 1

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

Problem 1 (Principle of Inclusion and Exclusion)

Let [math]\displaystyle{ n\ge 1 }[/math] be a positive integer and [math]\displaystyle{ A_1,A_2,\ldots,A_n }[/math] be [math]\displaystyle{ n }[/math] events.

• [Union bound] Prove [math]\displaystyle{ \mathbf{Pr}\left( \bigcup_{i=1}^{n} A_i \right) \le \sum_{i=1}^{n} \mathbf{Pr}\left(A_i\right) }[/math] using the definition of probability space.
• [Principle of Inclusion and Exclusion (PIE)] Prove that [math]\displaystyle{ \mathbf{Pr}\left( \bigcup_{i=1}^n A_i\right) = \sum_{\emptyset \neq S \subseteq [n]} (-1)^{|S|-1} \mathbf{Pr}\left( \bigcap_{i \in S} A_i \right) }[/math], where [math]\displaystyle{ [n]=\{1,2,\ldots,n\} }[/math].
• [Surjection] For positive integers [math]\displaystyle{ m\ge n }[/math], prove that the probability of a uniform random function [math]\displaystyle{ f:[m]\to[n] }[/math] to be surjective (满射) is [math]\displaystyle{ \sum_{k=1}^n(-1)^{n-k}{n\choose k}\left(\frac{k}{n}\right)^m }[/math].
• [Coprime integers] Given positive integers [math]\displaystyle{ n \ge 2 }[/math], calculate the number of integer pairs [math]\displaystyle{ (x,y) }[/math] satisfying [math]\displaystyle{ 1 \le x \lt y \le n }[/math] and [math]\displaystyle{ \mathrm{gcd}(x,y) = 1 }[/math]. (Remark: It suggests the probability that two randomly chosen integers from [math]\displaystyle{ 1 }[/math] to [math]\displaystyle{ n }[/math] are coprime tends to [math]\displaystyle{ \frac{6}{\pi^2} }[/math] as [math]\displaystyle{ n }[/math] tends to infinity.)
• [Bonferroni's inequality and Kounias' inequality] Prove that [math]\displaystyle{ \sum_{i=1}^n \mathbf{Pr}(A_i) - \sum_{1 \le i\lt j \le n} \mathbf{Pr}(A_i \cap A_j)\le \mathbf{Pr}\left(\bigcup_{i=1}^n A_i\right) \le \sum_{i=1}^n \mathbf{Pr} \left( A_i\right) - \sum_{i=2}^n \mathbf{Pr}(A_1 \cap A_i). }[/math] (Hint: This is sometimes called Kounias' inequality which is weaker than the Bonferroni's inequality. You can try using Venn diagram to understand these inequalities.)

Problem 2 (Symmetric 1D random walk)

A gambler plays a fair gambling game: At each round, he flips a fair coin, earns [math]\displaystyle{ 1 }[/math] point if it's HEADs, and loses [math]\displaystyle{ 1 }[/math] point if otherwise.

• [Symmetric 1D random walk (I)] Let [math]\displaystyle{ A_i }[/math] be the event that the gambler earns [math]\displaystyle{ 0 }[/math] points after playing [math]\displaystyle{ i }[/math] rounds of the game, that is, the number of times the coin lands on heads is equal to the number of times it lands on tails. Compute [math]\displaystyle{ \sum_{i=1}^{+\infty} \mathbf{Pr}(A_i) }[/math]. (Hint: You may use Stirling's approximation to estimate [math]\mathbf{Pr}(A_i)[/math] and derive that [math]\displaystyle{ \sum_{i=1}^{+\infty} \mathbf{Pr}(A_i) = +\infty }[/math].)

• [Symmetric 1D random walk (II)] Suppose that the game ends upon that the gambler loses all his [math]\displaystyle{ m }[/math] points. Let [math]\displaystyle{ B_i }[/math] be the event that the game ends within [math]\displaystyle{ i }[/math] rounds. Compute [math]\displaystyle{ \sum_{i=1}^{+\infty} \mathbf{Pr}(\overline{B_i}) }[/math]. (Hint: You may first consider [math] \displaystyle{m = 1}[/math] case. Let [math]\displaystyle{C_i}[/math] be the event that the game ends at the [math]\displaystyle{i}[/math]-th round. (i) Prove that [math]\displaystyle{ \sum_{i=1}^{+\infty} \mathbf{Pr}(\overline{B_i}) = \sum_{i=1}^{+\infty} (i-1) \mathbf{Pr}(C_i) }[/math]. (ii) Compute [math]\displaystyle{ \mathbf{Pr}(C_i) }[/math], which is a special case of the ballot problem. (iii) Finally, use Stirling's approximation to derive that [math]\displaystyle{ \sum_{i=1}^{+\infty} \mathbf{Pr}(\overline{B_i}) = +\infty }[/math].)

Problem 3 (Probability space)

• [Nonexistence of probability space] Prove that it is impossible to define a uniform probability law on natural numbers [math]\displaystyle{ \mathbb{N} }[/math]. More precisely, prove that there does not exist a probability space [math]\displaystyle{ (\mathbb{N},2^{\mathbb{N}},\mathbf{Pr}) }[/math] such that [math]\displaystyle{ \mathbf{Pr}(\{i\}) = \mathbf{Pr}(\{j\}) }[/math] for all [math]\displaystyle{ i, j \in \mathbb{N} }[/math]. Please explain why the same argument fails to prove that there is no uniform probability law on the real interval [math]\displaystyle{ [0,1] }[/math], that is, there is no such probability space [math]\displaystyle{ ([0,1],\mathcal{F},\mathbf{Pr}) }[/math] that for any interval [math]\displaystyle{ (l,r] \subseteq [0,1] }[/math], it holds that [math]\displaystyle{ (l,r] \in \mathcal{F} }[/math] and [math]\displaystyle{ \mathbf{Pr}( (l,r] ) = r-l }[/math]. (Actually, such probability measure does exist and is called the Lebesgue measure on [math]\displaystyle{ [0,1] }[/math]).

• [Smallest [math]\displaystyle{ \sigma }[/math]-field] For any subset [math]\displaystyle{ S \subseteq 2^\Omega }[/math], prove that the smallest [math]\displaystyle{ \sigma }[/math]-field containing [math]\displaystyle{ S }[/math] is given by [math]\displaystyle{ \bigcap_{\substack{S \subseteq \mathcal{F} \subseteq 2^\Omega\\ \mathcal{F} \text{ is a } \sigma\text{-field }} } \mathcal{F} }[/math]. (Hint: You should show that it is indeed a [math]\displaystyle{ \sigma }[/math]-field and also it is the smallest one containing [math]\displaystyle{ S }[/math].)

• [Union of [math]\displaystyle{ \sigma }[/math]-field] Let [math]\displaystyle{ \mathcal{F} }[/math] and [math]\displaystyle{ \mathcal{G} }[/math] be [math]\displaystyle{ \sigma }[/math]-fields of subsets of [math]\displaystyle{ \Omega }[/math]. Show that [math]\displaystyle{ \mathcal{F} \cup \mathcal{G} }[/math] is not necessarily a [math]\displaystyle{ \sigma }[/math]-field.

• [Probability space?] Let [math]\displaystyle{ \Omega = \mathbb{R} }[/math], [math]\displaystyle{ \mathcal{F} }[/math] is the set of all subsets [math]\displaystyle{ A \subseteq \Omega }[/math] so that [math]\displaystyle{ A }[/math] or [math]\displaystyle{ \overline{A} }[/math] (complement of [math]\displaystyle{ A }[/math]) is countable, [math]\displaystyle{ P(A) = 0 }[/math] in the first case and [math]\displaystyle{ P(A) = 1 }[/math] in the second. Is [math]\displaystyle{ (\Omega,\mathcal{F},P) }[/math] a probability space? Please explain your answer.

• [[math]\displaystyle{ \sigma }[/math]-field?] A set [math]\displaystyle{ A \subseteq \mathbb{N} }[/math] is said to have asymptotic density [math]\displaystyle{ \theta }[/math] if [math]\displaystyle{ \lim_{n \to \infty} |A \cap \{1,2,\ldots,n\}| / n = \theta }[/math]. Let [math]\displaystyle{ \mathcal{A} }[/math] be the collection of sets for which the asymptotic density exists. Is [math]\displaystyle{ \mathcal{A} }[/math] a [math]\displaystyle{ \sigma }[/math]-algebra? Please explain your answer.

Problem 4 (Conditional probability)

• [Positive correlation] We say that events [math]\displaystyle{ B }[/math] gives positive information about event [math]\displaystyle{ A }[/math] if [math]\displaystyle{ \mathbf{Pr}(A|B) \gt \mathbf{Pr}(A) }[/math], that is, the occurrence of [math]\displaystyle{ B }[/math] makes the occurrence of [math]\displaystyle{ A }[/math] more likely. Now suppose that [math]\displaystyle{ B }[/math] gives positive information about [math]\displaystyle{ A }[/math].
  1. Does [math]\displaystyle{ A }[/math] give positive information about [math]\displaystyle{ B }[/math]?
  2. Does [math]\displaystyle{ \overline{B} }[/math] give negative information about [math]\displaystyle{ A }[/math], that is, is it true that [math]\displaystyle{ \mathbf{Pr}(A|\overline{B}) \lt \mathbf{Pr}(A) }[/math]?
  3. Does [math]\displaystyle{ \overline{B} }[/math] give positive information or negative information about [math]\displaystyle{ \overline{A} }[/math]?

• [Balls in urns (I)] There are [math]\displaystyle{ n }[/math] urns of which the [math]\displaystyle{ r }[/math]-th contains [math]\displaystyle{ r-1 }[/math] white balls and [math]\displaystyle{ n-r }[/math] black balls. You pick an urn uniformly at random (here, "uniformly" means that each urn has equal probability of being chosen) and remove two balls from that urn, uniformly at random without replacement (which means that each of the [math]\displaystyle{ {n-1\choose 2} }[/math] pairs of balls are chosen to be removed with equal probability). Find the following probabilities:

  1. the second ball is black;
  2. the second ball is black, given that the first is black.

• [Balls in urns (II)] Suppose that an urn contains [math]\displaystyle{ w }[/math] white balls and [math]\displaystyle{ b }[/math] black balls. The balls are drawn from the urn one by one, each time uniformly and independently at random, without replacement (which means we do not put the chosen ball back after each drawing). Find the probabilities of the events:

  1. the first white ball drawn is the [math]\displaystyle{ (k+1) }[/math]th ball;
  2. the last ball drawn is white.

• [Noisy channel] When coded messages are sent, there are sometimes errors in transmission. In particular, Morse code uses "dots" and "dashes", which are known to occur in the proportion of 3:4. In other words, the probability of a "dot" being sent is [math]\displaystyle{ 3/7 }[/math], and the probability of a "dash" being sent is [math]\displaystyle{ 4/7 }[/math]. Suppose due to interference on the transmission channel, with probability [math]\displaystyle{ 1/8 }[/math], a dot will be mistakenly received as a dash, and vice versa. If we receive a dot, what is the probability that the transmitted symbol is indeed a dot?

Problem 5 (Independence)

Let's consider a series of [math]\displaystyle{ n }[/math] outputs [math]\displaystyle{ (X_1, X_2, \cdots, X_n) \in \{0,1\}^n }[/math] of [math]\displaystyle{ n }[/math] independent Bernoulli trials, where each trial succeeds with the same probability [math]\displaystyle{ 0 \lt p \lt 1 }[/math].

• [Limited independence] Construct three events [math]\displaystyle{ A,B }[/math] and [math]\displaystyle{ C }[/math] out of [math]\displaystyle{ n }[/math] Bernoulli trials such that [math]\displaystyle{ A, B }[/math] and [math]\displaystyle{ C }[/math] are pairwise independent but are not (mutually) independent. You need to prove that the constructed events [math]\displaystyle{ A, B }[/math] and [math]\displaystyle{ C }[/math] satisfy this. (Hint: Consider the case where [math]\displaystyle{ n = 2 }[/math] and [math]\displaystyle{ p = 1/2 }[/math].)

• [Product distribution] Suppose someone has observed the output of the [math]\displaystyle{ n }[/math] trials, and she told you that precisely [math]\displaystyle{ k }[/math] out of [math]\displaystyle{ n }[/math] trials succeeded for some [math]\displaystyle{ 0\lt k\lt n }[/math]. Now you want to predict the output of the [math]\displaystyle{ (n+1) }[/math]-th trial while the parameter [math]\displaystyle{ p }[/math] of the Bernoulli trial is unknown. One way to estimate [math]\displaystyle{ p }[/math] is to find such [math]\displaystyle{ \hat{p} }[/math] that makes the observed outcomes most probable, namely you need to solve [math]\displaystyle{ \arg \max_{\hat{p}\in(0,1)} \mathbf{Pr}_{\hat{p}} [k \text{ out of } n\text{ trials succeed}]. }[/math]

  1. Estimate [math]\displaystyle{ p }[/math] by solving the above optimization problem.
  2. If someone tells you exactly which [math]\displaystyle{ k }[/math] trials succeed (in addition to just telling you the number of successful trials, which is [math]\displaystyle{ k }[/math]), would it help you to estimate [math]\displaystyle{ p }[/math] more accurately? Why?

Problem 6 (Probabilistic method)

A CNF formula （合取范式） [math]\displaystyle{ \Phi }[/math] over [math]\displaystyle{ n }[/math] Boolean variables [math]\displaystyle{ x_1,\cdots, x_n }[/math] is a conjunction ([math]\displaystyle{ \land }[/math]) of clauses (子句) [math]\displaystyle{ \Phi=C_1\land C_2\land\cdots\land C_m }[/math], where each clause [math]\displaystyle{ C_j=\ell_{j_1}\lor\ell_{j_2}\cdots\lor\ell_{j_k} }[/math] is a disjunction ([math]\displaystyle{ \lor }[/math]) of literals (文字), where a literal [math]\displaystyle{ \ell_r }[/math] is either a variable [math]\displaystyle{ x_i }[/math] or the negation [math]\displaystyle{ \bar{x}_i }[/math] of a variable. A CNF formula is satisfiable (可满足) if there is a truth assignment [math]\displaystyle{ x=(x_1,\cdots, x_n)\in \{\mathtt{true},\mathtt{false}\}^n }[/math] to the variables such that [math]\displaystyle{ \Phi(x)=\mathtt{true} }[/math]. A [math]\displaystyle{ k }[/math]-CNF formula is a CNF formula in which each clause contains exactly [math]\displaystyle{ k }[/math] literals (without repetition).

• [Satisfiability (I)] Let [math]\displaystyle{ \Phi }[/math] be a [math]\displaystyle{ k }[/math]-CNF with less than [math]\displaystyle{ 2^k }[/math] clauses. Use the probabilistic method to show that [math]\displaystyle{ \Phi }[/math] must be satisfiable. You should be explicit about the probability space that is used.
• [Satisfiability (II)] Give a constructive proof of the same problem above. That is, prove that [math]\displaystyle{ \Phi }[/math] is satisfiable by showing how to construct a truth assignment [math]\displaystyle{ x=(x_1,\cdots, x_n)\in \{\mathtt{true},\mathtt{false}\}^n }[/math] such that [math]\displaystyle{ \Phi(x)=\mathtt{true} }[/math]. Your construction does NOT have to be efficient. Please explain the difference between this constructive proof and the proof by the probabilistic method.
• [Satisfiability (III)] Let [math]\displaystyle{ \Phi }[/math] be a [math]\displaystyle{ k }[/math]-CNF with [math]\displaystyle{ m\geq 2^k }[/math] clauses. Use the probabilistic method to show that there exists a truth assignment [math]\displaystyle{ x=(x_1,\cdots, x_n)\in \{\mathtt{true},\mathtt{false}\}^n }[/math] satisfying at least [math]\displaystyle{ \lfloor m(1-1/2^k) \rfloor }[/math] clauses in [math]\displaystyle{ \Phi }[/math]. (Hint: Consider overlaps of events in Venn diagram.) You should be explicit about the probability space that is used.