A smooth circular wire of radius \(a\) is held fixed in a vertical plane with light elastic strings of natural length \(a\) and modulus \(\lambda\) attached to the upper and lower extremities, \(A\) and \(C\) respectively, of the vertical diameter. The other ends of the two strings are attached to a small ring \(B\) which is free to slide on the wire. Show that, while both strings remain taut, the equation for the motion of the ring is $$2ma \ddot\theta=\lambda(\cos\theta-\sin\theta)-mg\sin\theta,$$ where \(\theta\) is the angle \( \angle{CAB}\). Initially the system is at rest in equilibrium with \(\sin\theta=\frac{3}{5}\). Deduce that \(5\lambda=24mg\). The ring is now displaced slightly. Show that, in the ensuing motion, it will oscillate with period approximately $$10\pi\sqrt{a\over91g}\,.$$
It has been observed that Professor Ecks proves three types of theorems: 1, those that are correct and new; 2, those that are correct, but already known; 3, those that are false. It has also been observed that, if a certain of her theorems is of type \(i\), then her next theorem is of type \(j\) with probability \(p\low_{ij},\) where \(p\low_{ij}\) is the entry in the \(i\)th row and \(j\)th column of the following array: \[ \begin{pmatrix}0.3 & 0.3 & 0.4\\ 0.2 & 0.4 & 0.4\\ 0.1 & 0.3 & 0.6 \end{pmatrix}\,. \] Let \(a_{i},\) \(i=1,2,3\), be the probability that a given theorem is of type \(i\), and let \(b_{j}\) be the consequent probability that the next theorem is of type \(j\).
Let \(X\) be a random variable which takes only the finite number of different possible real values \(x_{1},x_{2},\ldots,x_{n}.\) Define the expectation \(\mathbb{E}(X)\) and the variance \(\var(X)\) of \(X\). Show that , if \(a\) and \(b\) are real numbers, then \(\E(aX+b)=a\E(X)+b\) and express \(\var(aX+b)\) similarly in terms of \(\var(X)\). Let \(\lambda\) be a positive real number. By considering the contribution to \(\var(X)\) of those \(x_{i}\) for which \(\left|x_{i}-\E(X)\right|\geqslant\lambda,\) or otherwise, show that \[ \mathrm{P}\left(\left|X-\E(X)\right|\geqslant\lambda\right)\leqslant\frac{\var(X)}{\lambda^{2}}\,. \] Let \(k\) be a real number satisfying \(k\geqslant\lambda.\) If \(\left|x_{i}-\E(X)\right|\leqslant k\) for all \(i\), show that \[ \mathrm{P}\left(\left|X-\E(X)\right|\geqslant\lambda\right)\geqslant\frac{\var(X)-\lambda^{2}}{k^{2}-\lambda^{2}}\,. \]
Solution: Definition: \(\displaystyle \mathbb{E}(X) = \sum_{i=1}^n x_i \mathbb{P}(X = x_i)\) Definition: \(\displaystyle \mathrm{Var}(X) = \sum_{i=1}^n (x_i-\mathbb{E}(X))^2 \mathbb{P}(X = x_i)\) Claim: \(\mathbb{E}(aX+b) = a\mathbb{E}(X)+b\) Proof: \begin{align*} \mathbb{E}(aX+b) &= \sum_{i=1}^n (ax_i+b) \mathbb{P}(X = x_i) \\ &= a\sum_{i=1}^n x_i \mathbb{P}(X = x_i) + b\sum_{i=1}^n \mathbb{P}(X = x_i)\\ &= a \mathbb{E}(X) + b \end{align*} Claim: \(\mathrm{Var}(aX+b) = a^2 \mathrm{Var}(X)\) Claim: \(\mathrm{P}\left(\left|X-\mathrm{E}(X)\right|\geqslant\lambda\right)\leqslant\frac{\mathrm{var}(X)}{\lambda^{2}}\) Proof: \begin{align*} \mathrm{Var}(X) &= \sum_{i=1}^n (x_i-\mathbb{E}(X))^2 \mathbb{P}(X = x_i) \\ &\geq \sum_{|x_i - \mathbb{E}(X)| \geq \lambda} (x_i-\mathbb{E}(X))^2 \mathbb{P}(X = x_i) \\ &\geq \sum_{|x_i - \mathbb{E}(X)| \geq \lambda} \lambda^2 \mathbb{P}(X = x_i) \\ &= \lambda^2 \sum_{|x_i - \mathbb{E}(X)| \geq \lambda} \mathbb{P}(X = x_i) \\ &= \lambda^2 \mathrm{P}\left(\left|X-\mathrm{E}(X)\right|\geqslant\lambda\right) \end{align*} Claim: \[ \mathrm{P}\left(\left|X-\mathrm{E}(X)\right|\geqslant\lambda\right)\geqslant\frac{\mathrm{var}(X)-\lambda^{2}}{k^{2}-\lambda^{2}}\,. \] Proof: \begin{align*} && \mathrm{Var}(X) &= \sum_{i=1}^n (x_i-\mathbb{E}(X))^2 \mathbb{P}(X = x_i) \\ &&&= \sum_{|x_i - \mathbb{E}(X)| \geq \lambda} (x_i-\mathbb{E}(X))^2 \mathbb{P}(X = x_i) + \sum_{|x_i - \mathbb{E}(X)| < \lambda} (x_i-\mathbb{E}(X))^2 \mathbb{P}(X = x_i) \\ &&& \leq \sum_{|x_i - \mathbb{E}(X)| \geq \lambda} k^2 \mathbb{P}(X = x_i) + \sum_{|x_i - \mathbb{E}(X)| < \lambda} \lambda^2 \mathbb{P}(X = x_i) \\ &&&= k^2 \mathbb{P}\left(\left|X-\mathrm{E}(X)\right|\geqslant\lambda\right) + \lambda^2 \mathbb{P}\left(\left|X-\mathrm{E}(X)\right| < \lambda\right) \\ &&&= k^2 \mathbb{P}\left(\left|X-\mathrm{E}(X)\right|\geqslant\lambda\right) + \lambda^2(1- \mathbb{P}\left(\left|X-\mathrm{E}(X)\right| \leq \lambda\right) \\ &&&= (k^2 - \lambda^2) \mathbb{P}\left(\left|X-\mathrm{E}(X)\right|\geqslant\lambda\right) + \lambda^2 \\ \Rightarrow&& \frac{\mathrm{Var}(X)-\lambda^2}{k^2 - \lambda^2} &\leq \mathbb{P}\left(\left|X-\mathrm{E}(X)\right|\geqslant\lambda\right) \end{align*} [Note: This result is known as Chebyshev's inequality, and is an important starting point to understanding the behaviour of tails of random variables]
Whenever I go cycling I start with my bike in good working order. However if all is well at time \(t\), the probability that I get a puncture in the small interval \((t,t+\delta t)\) is \(\alpha\,\delta t.\) How many punctures can I expect to get on a journey during which my total cycling time is \(T\)? When I get a puncture I stop immediately to repair it and the probability that, if I am repairing it at time \(t\), the repair will be completed in time \((t,t+\delta t)\) is \(\beta\,\delta t.\) If \(p(t)\) is the probability that I am repairing a puncture at time \(t\), write down an equation relating \(p(t)\) to \(p(t+\delta t)\), and derive from this a differential equation relating \(p'(t)\) and \(p(t).\) Show that \[ p(t)=\frac{\alpha}{\alpha+\beta}(1-\mathrm{e}^{-(\alpha+\beta)t}) \] satisfies this differential equation with the appropriate initial condition. Find an expression, involving \(\alpha,\beta\) and \(T\), for the time expected to be spent mending punctures during a journey of total time \(T\). Hence, or otherwise, show that, the fraction of the journey expected to be spent mending punctures is given approximately by \[ \quad\frac{\alpha T}{2}\quad\ \mbox{ if }(\alpha+\beta)T\text{ is small, } \] and by \[ \frac{\alpha}{\alpha+\beta}\quad\mbox{ if }(\alpha+\beta)T\text{ is large.} \]