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Title: Conditional Expectation (given events)
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Series: Probability Theory
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Chapter: Random Variables
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YouTube-Title: Probability Theory 21 | Conditional Expectation (given events)
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Subtitle on GitHub: pt21_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 $B \in \mathcal{A}$ an event with $\mathbb{P}(B) > 0$. For a random variable $X : \Omega \rightarrow \mathbb{R}$, we can define the conditional expectation $\mathbb{E}(X|B)$ if it exists. What is not correct?
A1: $\mathbb{E}(X|B)$ is a real number.
A2: $ \displaystyle \mathbb{E}(X|B) = \frac{1}{\mathbb{P}(B)} \mathbb{E}( \mathbf{1}_B X ) $
A3: $ \displaystyle \mathbb{E}(X|B) = \frac{1}{\mathbb{P}(B)}\int_B X , d \mathbb{P} $
A4: $ \displaystyle \mathbb{E}(X|B) = \int_{\Omega}( X - \mathbf{1}_B ) , d \mathbb{P} $
Q2: What is the conditional expectation of $\mathbb{E}(\mathbf{1}_B|B)$ for an event with $\mathbb{P}(B) > 0$.
A1: $1$
A2: $0$
A3: $\frac{1}{2}$
A4: $-1$
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Last update: 2025-09
