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: $$\mathrm{Var}(X) =\int_{\mathbb{R}} (x - \mathbb{E}(X))^2 f_X(x) , dx $$

A2: $$\mathrm{Var}(X) =\int_{\mathbb{R}} x^2 f_X(x) , dx $$

A3: $$\mathrm{Var}(X) =\int_{\mathbb{R}} (x - \mathbb{E}(X)^2) f_X(x) , dx $$

A4: $$\mathrm{Var}(X) =\int_{\mathbb{R}} (x^2 - \mathbb{E}(X)^2 ) f_X(x) , dx $$

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: $$\mathrm{Var}(X) =\sum_{x \in \mathbb{R}} (x - \mathbb{E}(X))^2 \mathbb{P}_X( { x } ) $$

A2: $$\mathrm{Var}(X) =\sum_{x \in \mathbb{R}} x^2 \mathbb{P}_X( { x } ) $$

A3: $$\mathrm{Var}(X) =\sum_{x \in \mathbb{R}} (x^2 - \mathbb{E}(X) )\mathbb{P}_X( { x } ) $$

A4: $$\mathrm{Var}(X) =\sum_{x \in \mathbb{R}} \mathbb{E}(X)^2 \mathbb{P}_X( { x } ) $$