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Title: Weak Law of Large Numbers
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Series: Probability Theory
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Chapter: Convergence of Random Variables
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YouTube-Title: Probability Theory 28 | Weak Law of Large Numbers
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Bright video: Watch on YouTube
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Dark video: Watch on YouTube
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Ad-free video: Watch Vimeo video
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: pt28_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: What does it mean that a family of random variables is i.i.d?
A1: It means that they are independent and identically distributed.
A2: It means that they are independent and identically defined.
A3: It means that they are identically distributed and intelligent.
A4: It means that they are intrinsically independent distributed.
Q2: Let $X_k: \Omega \rightarrow \mathbb{R}$ be a random variable where $\mathbb{E}(|X_1|)$ exists. What is the correct statement for the weak law of large numbers?
A1: If $(X_k){k \in \mathbb{N}}$ are i.i.d., then \newline $ \displaystyle \mathbb{P}\left( \Big| \frac{1}{n} \sum{k = 1}^n X_k - \mathbb{E}(X_1) \Big| \geq \varepsilon \right) $ $ \displaystyle \xrightarrow{n \rightarrow \infty} 0 $
A2: If $(X_k){k \in \mathbb{N}}$ are i.i.d., then $ \displaystyle\mathbb{P}\left( \Big| \frac{1}{n} \sum{k = 1}^n X_k - \mathbb{E}(X_1) \Big| = \varepsilon \right) $ $ \displaystyle \xrightarrow{n \rightarrow \infty} 0 $
A3: If $(X_k){k \in \mathbb{N}}$ are i.i.d., then $ \displaystyle \mathbb{P}\left( \Big| \frac{1}{n} \sum{k = 1}^n X_k - \mathbb{E}(X_1) \Big| \leq \varepsilon \right) $ $ \displaystyle \xrightarrow{n \rightarrow \infty} 0 $
A4: If $(X_k){k \in \mathbb{N}}$ are i.i.d., then $ \displaystyle \mathbb{P}\left( \Big| \frac{1}{n} \sum{k = 1}^n X_k - \mathbb{E}(X_1) \Big| \leq \varepsilon \right) $ $ \displaystyle \xrightarrow{n \rightarrow \infty} \frac{1}{\varepsilon} $
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Last update: 2025-09
