# 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 $$Z$$?
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 $$(\mathcal{F}_n)_{n \geq 0}$$.