OSA: Probability Theory

Question 1

Let \(X\) be a continuously distributed real-valued random variable with probability density function \[ \rho(x) = \begin{cases} 6x - 6x^2 & \text{if }0 \leq x \leq 1\text{ and} \\ 0 & \text{otherwise}, \end{cases} \] and let \(Z := \sqrt{20}\cdot(X-\frac{1}{2})\).


What is the expected value of \(Z\)?

What is the variance of \(Z\)?

Question 2

Let \((X_n)_{n \in \mathbb{N}}\) be independent integrable random variables, \(\mathcal{F}_n = \sigma(X_i : i \leq n)\), \(\mathcal{F}_0 = \{\emptyset, \Omega\}\). Consider \[ S_n := \sum_{i=1}^n X_i. \] Determine \(a\) such that the condition \[ \mathbb{E}(X_n) \geq a\text{ for all }n \in \mathbb{N} \] is equivalent to \((S_n)_{n \in \mathbb{N}}\) being a submartingale with respect to the filtration \((\mathcal{F}_n)_{n \geq 0}\).