数据科学基础 (Fall 2024)/Problem Set 1

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Problem 1 (Probability space)

• [De Morgan’s Laws] Let [math]\displaystyle{ \{A_i\} }[/math] be a collection of sets. Prove [math]\displaystyle{ \left(\bigcup_iA_i\right)^c=\bigcap_iA_i^c }[/math].
• [[math]\displaystyle{ \sigma }[/math]-algebra] Let [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] belong to some [math]\displaystyle{ \sigma }[/math]-algebra [math]\displaystyle{ \mathcal F }[/math]. Show that [math]\displaystyle{ \mathcal F }[/math] contains the sets [math]\displaystyle{ A\cap B }[/math], [math]\displaystyle{ A\setminus B }[/math], and [math]\displaystyle{ A\oplus B }[/math], but unnecessarily contains the set [math]\displaystyle{ A\cup B }[/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].)

• [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].

• [Basic properties] Prove the following basic properties of probability:

  • [math]\displaystyle{ \Pr(A^c)=1-\Pr(A) }[/math]
  • [math]\displaystyle{ \Pr(\emptyset)=0 }[/math]
  • [math]\displaystyle{ \Pr(A\setminus B)=\Pr(A)-\Pr(A\cap B) }[/math]
  • [math]\displaystyle{ A\subseteq B\implies \Pr(A)\le \Pr(B) }[/math]
  • [math]\displaystyle{ \Pr(A\cup B)=\Pr(A)+\Pr(B)-\Pr(A\cap B) }[/math]
  • [math]\displaystyle{ \Pr(A\oplus B)=\Pr(A)+\Pr(B)-2\Pr(A\cap B) }[/math]

• [Murphy’s Law] A fair coin is tossed repeatedly. Show that, with probability one, a head turns up sooner or later. Show similarly that any given finite sequence of heads and tails occurs eventually with probability one. Explain the connection with Murphy’s Law.