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 \(Z := \sqrt{20}\cdot(X-\frac{1}{2})\).

question	answer	result	explanation
What is the expected value of \(Z\)?			Notice that \(\rho(x) = \rho(1-x)\), so the distribution of \(X\) is symmetric about \(\frac{1}{2}\) and therefore the distribution of \(Z\) is symmetric about \(0\).
What is the variance of \(Z\)?			By the first part of the question we know that \(\mathbb{E}Z = 0\), so the variance is \(\mathbb{E}(Z-\mathbb{E}Z)^2 = \mathbb{E}Z^2\), for which we get \[ \begin{split} \mathbb{E}Z^2 &= \int_0^1 \\ \big(\sqrt{20}(x-\frac{1}{2})\big)^2(6x-6x^2)\,\mathrm{d}x &= 120\cdot\int_0^1 (x^2-x+\frac{1}{4})x(1-x)\,\mathrm{d}x \\ &= 120\cdot\int_0^1 (-x^4 + 2x^3 - \frac{5}{4}x^2 + \frac{1}{4}x\,\mathrm{d}x \\ &= 120\cdot(-\frac{1}{5} + \frac{1}{2} - \frac{5}{12} + \frac{1}{8}) \\ &= -24 + 60 - 50 + 15 = 1 \end{split} \]

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 \((\mathcal{F}_n)_{n \geq 0}\).

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