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Title: Expectation and Change-of-Variables
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Series: Probability Theory
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YouTube-Title: Probability Theory 14 | Expectation and Change-of-Variables
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Bright video: https://youtu.be/EhVbqe8J_Ww
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Dark video: https://youtu.be/rZtltvKNA2A
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Ad-free video: Watch Vimeo video
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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: pt14_sub_eng.srt missing
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Definitions in the video: expectation of a random variable
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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 expectation $\mathbb{E}(X)$?
A1: $\mathbb{E}(X) \in \mathbb{R}$
A2: $\mathbb{E}(X) = \int_{\Omega} X , d\mathbb{P}$
A3: $\mathbb{E}(X) = \int_{\Omega} X(\omega) , d\mathbb{P}(\omega)$
A4: $\mathbb{E}(X) = \int_{X(\Omega)} x , d\mathbb{P}_X(x)$
A5: $\mathbb{E}(X) = \int_{X(\Omega)} , d\mathbb{P}_X(x)$
Q2: Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X \colon \Omega \rightarrow \mathbb{R}$ be a random variable with a discrete distribution. Which statement is correct?
A1: $\mathbb{E}(X) = \sum_{j=1}^\infty j^2 $
A2: $\mathbb{E}(X) = \sum_{x \in \mathbb{R}} \mathbb{P}(X = x)$
A3: $\mathbb{E}(X) = \sum_{x \in \mathbb{R}} x \cdot \mathbb{P}(X = x)$
A4: $\mathbb{E}(X) = \sum_{x \in \mathbb{R}} x^2 \cdot \mathbb{P}(X = x)$
Q3: Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X \colon \Omega \rightarrow \mathbb{R}$ be a random variable with a continuous distribution given by a pdf $f_X$. Which statement is correct?
A1: $\mathbb{E}(X) = \int_{\mathbb{R}} x , dx$
A2: $\mathbb{E}(X) = \int_{\mathbb{R}} x f_X(x) , dx$
A3: $\mathbb{E}(X) = \int_{\mathbb{R}} x^2 f_X(x) , dx$
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Last update: 2024-10
