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Title: Variance
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Series: Probability Theory
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Chapter: Random Variables
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YouTube-Title: Probability Theory 16 | Variance
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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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Exercise Download PDF sheets
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: pt16_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: Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X \colon \Omega \rightarrow \mathbb{R}$ be a random variable. What is not correct for the variance $\mathrm{Var}(X)$?
A1: $\mathrm{Var}(X) $ $ = \mathbb{E}( (X-\mathbb{E}(X))^2 )$
A2: $\mathrm{Var}(X) $ $ = \mathbb{E}( X^2 ) - \mathbb{E}( X )^2$
A3: $\mathrm{Var}(X) $ $ = \mathbb{E}( X )^2 - \mathbb{E}( X^2 )$
A4: $\mathrm{Var}(X) $ $ = \mathbb{E} \left( X^2 - 2 \mathbb{E}( X ) X + \mathbb{E}( X )^2 \right) $
Q2: Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X \colon \Omega \rightarrow \mathbb{R}$ be a continuous random variable with pdf $f_X$. What is correct for the variance $\mathrm{Var}(X)$?
A1: $\displaystyle \mathrm{Var}(X) $ $\displaystyle = \int_{\mathbb{R}} (x - \mathbb{E}(X))^2 f_X(x) , dx $
A2: $\displaystyle \mathrm{Var}(X) $ $\displaystyle = \int_{\mathbb{R}} x^2 f_X(x) , dx $
A3: $\displaystyle \mathrm{Var}(X) $ $\displaystyle = \int_{\mathbb{R}} (x - \mathbb{E}(X)^2) f_X(x) , dx $
A4: $\displaystyle \mathrm{Var}(X) $ $\displaystyle = \int_{\mathbb{R}} (x^2 - \mathbb{E}(X)^2 ) f_X(x) , dx $
Q3: Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X \colon \Omega \rightarrow \mathbb{R}$ be a discrete random variable. What is correct for the variance $\mathrm{Var}(X)$?
A1: $ \displaystyle \mathrm{Var}(X) $ $ \displaystyle = \sum_{x \in \mathbb{R}} (x - \mathbb{E}(X))^2 \mathbb{P}_X( { x } ) $
A2: $\displaystyle \mathrm{Var}(X) $ $\displaystyle = \sum_{x \in \mathbb{R}} x^2 \mathbb{P}_X( { x } ) $
A3: $\displaystyle \mathrm{Var}(X) $ $\displaystyle = \sum_{x \in \mathbb{R}} (x^2 - \mathbb{E}(X) )\mathbb{P}_X( { x } ) $
A4: $\displaystyle \mathrm{Var}(X) $ $\displaystyle = \sum_{x \in \mathbb{R}} \mathbb{E}(X)^2 \mathbb{P}_X( { x } ) $
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Last update: 2025-09
