Q1: Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X \colon \Omega \rightarrow \mathbb{R}$ be a random variable with $X \sim \mathrm{Exp}(1)$. What is a correct pdf for $\mathbb{P}_X$?

A1: $$f_X(x) = \begin{cases} -e^{-x} , , & x \geq 0 \ e^{-x} , , & x < 0\end{cases}$$

A2: $$f_X(x) = \begin{cases} e^{-x} , , & x > 0 \ 0 , , & x \leq 0\end{cases}$$

A3: $$f_X(x) = \begin{cases} -e^{-x} , , & x > 0 \ 0 , , & x \leq 0\end{cases}$$

A4: $$f_X(x) = \begin{cases} -e^{-x} , , & x > 0 \ 1 , , & x \leq 0\end{cases}$$

Q2: Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X \colon \Omega \rightarrow \mathbb{R}$ be a random variable with $X \sim \mathrm{Exp}(1)$. What is $\mathbb{E}(X)$?

A1: $1$

A2: $\frac{1}{2}$

A3: $-\frac{1}{2}$

A4: $\frac{1}{4}$

Q3: Let $X \sim \mathrm{Exp}(2)$ and $Y \sim \mathrm{Exp}(3)$ be two independent random variables. What is $\mathbb{E}(X \cdot Y )$?

A1: $\frac{1}{6}$

A2: $\frac{1}{5}$

A3: $-\frac{1}{2}$

A4: $\frac{1}{4}$

A5: $\frac{1}{3}$