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Title: Independence for Random Variables
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Series: Probability Theory
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YouTube-Title: Probability Theory 13 | Independence for Random Variables
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Bright video: https://youtu.be/lawe9TT1p18
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Dark video: https://youtu.be/If_iTaWYeww
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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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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: pt13_sub_eng.srt missing
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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 $X \colon \Omega \rightarrow \mathbb{R}$, $Y \colon \Omega \rightarrow \mathbb{R}$ be two random variables. What is not a correct definition for the independence of $X$ and $Y$?
A1: $$\mathbb{P}(X \leq x, Y \leq y) =\mathbb{P}(X \leq x) \cdot \mathbb{P}(Y \leq y) $$ for all $x,y \in \mathbb{R}$.
A2: ${ \omega \in \Omega \mid X(\omega) \leq x }$ and ${ \omega \in \Omega \mid Y(\omega) \leq y }$ are independent events for all $x,y \in \mathbb{R}$.
A3: $F_{(X,Y)}(x,y) \leq F_X(x) \cdot F_Y(y)$ for all $x,y \in \mathbb{R}$.
A4: $\mathbb{P}(X \leq x, Y \leq y) = F_X(x) \cdot F_Y(y)$ for all $x,y \in \mathbb{R}$.
A5: $X^{-1}( (-\infty, x] )$ and $Y^{-1}( (-\infty, y] )$ are independent events for all $x,y \in \mathbb{R}$.
Q2: Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X \colon \Omega \rightarrow \mathbb{R}$, $Y \colon \Omega \rightarrow \mathbb{R}$ be two random variables. What is the cumulative distribution function of the random vector $(X,Y): \Omega \rightarrow \mathbb{R}^2$.
A1: $\mathbb{P}(X \leq x \text{ and } Y \leq y)$
A2: $\mathbb{P}(X \leq x \text{ or } Y \leq y)$
A3: $\mathbb{P}(X \leq x \text{ and } X \leq y)$
A4: $\mathbb{P}(X \leq x \text{ or } X \leq y)$
Q3: Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X_i \colon \Omega \rightarrow \mathbb{R}$ be a random variable for every $i \in \mathbb{N}$. What is the correct definition for the independence of the family of $(X_i)_{i \in \mathbb{N}}$?
A1: $$\mathbb{P}( (X_j \geq x_j){j \in J} ) = \prod{j \in J} \mathbb{P}(X_j \leq x_j)$$ for all finite $J \subseteq I$ and $x_j \in \mathbb{R}$.
A2: $$\mathbb{P}( (X_j \leq x_j){j \in J} ) = \prod{j \in J} \mathbb{P}(X_j \geq x_j)$$ for all finite $J \subseteq I$ and $x_j \in \mathbb{R}$.
A3: $$\mathbb{P}( (X_j \leq x_j){j \in J} ) = \prod{j \in J} \mathbb{P}(X_j \leq x_j)$$ for all finite $J \subseteq I$ and $x_j \in \mathbb{R}$.
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Last update: 2024-10
