Q1: What does it mean that a family of random variables is i.i.d?

A1: It means that they are independent and identically distributed.

A2: It means that they are independent and identically defined.

A3: It means that they are identically distributed and intelligent.

A4: It means that they are intrinsically independent distributed.

Q2: Let $X_k: \Omega \rightarrow \mathbb{R}$ be a random variable where $\mathbb{E}(|X_1|)$ exists. What is the correct statement for the weak law of large numbers?

A1: If $(X_k){k \in \mathbb{N}}$ are i.i.d., then $$ \mathbb{P}\left( \Big| \frac{1}{n} \sum{k = 1}^n X_k - \mathbb{E}(X_1) \Big| \geq \varepsilon \right) \xrightarrow{n \rightarrow \infty} 0 $$

A2: If $(X_k){k \in \mathbb{N}}$ are i.i.d., then $$ \mathbb{P}\left( \Big| \frac{1}{n} \sum{k = 1}^n X_k - \mathbb{E}(X_1) \Big| = \varepsilon \right) \xrightarrow{n \rightarrow \infty} 0 $$

A3: If $(X_k){k \in \mathbb{N}}$ are i.i.d., then $$ \mathbb{P}\left( \Big| \frac{1}{n} \sum{k = 1}^n X_k - \mathbb{E}(X_1) \Big| \leq \varepsilon \right) \xrightarrow{n \rightarrow \infty} 0 $$

A4: If $(X_k){k \in \mathbb{N}}$ are i.i.d., then $$ \mathbb{P}\left( \Big| \frac{1}{n} \sum{k = 1}^n X_k - \mathbb{E}(X_1) \Big| \leq \varepsilon \right) \xrightarrow{n \rightarrow \infty} \frac{1}{\varepsilon} $$