A circular wheel of radius \(r\) has moment of inertia \(I\) about its axle, which is fixed in a horizontal position. A light string is wrapped around the circumference of the wheel and a particle of mass \(m\) hangs from the free end. The system is released from rest and the particle descends. The string does not slip on the wheel. As the particle descends, the wheel turns through \(n_1\) revolutions, and the string then detaches from the wheel. At this moment, the angular speed of the wheel is \(\omega_0\). The wheel then turns through a further \(n_2\) revolutions, in time \(T\), before coming to rest. The couple on the wheel due to resistance is constant. Show that \[ \frac12 \omega_0 T = 2 \pi n_2\] and \[ I =\dfrac {mgrn_1T^2 -4\pi mr^2n_2^2}{4\pi n_2(n_1+n_2)}\;. \]
Let \(X\) be a random variable with a Laplace distribution, so that its probability density function is given by \[ \f(x) = \frac12 \e^{-\vert x \vert }\;, \text{ \ \ \ \(-\infty < x < \infty \)\ }. \tag{\(*\)} \] Sketch \(\f(x)\). Show that its moment generating function \({\rm M}_X(\theta)\) is given by ${\rm M}_X(\theta) = (1-\theta^2)^{-1}\( and hence find the variance of \)X$. A frog is jumping up and down, attempting to land on the same spot each time. In fact, in each of \(n\) successive jumps he always lands on a fixed straight line but when he lands from the \(i\)th jump (\(i=1\,,2\,,\ldots\,,n\)) his displacement from the point from which he jumped is \(X_i\,\)cm, where \(X_i\) has the distribution \((*)\). His displacement from his starting point after \(n\)~jumps is \(Y\,\)cm (so that \(Y=\sum\limits_{i=1}^n X_i\)). Each jump is independent of the others. Obtain the moment generating function for \(Y/ \sqrt {2n}\) and, by considering its logarithm, show that this moment generating function tends to \(\exp(\frac12\theta^2)\) as \(n\to\infty\). Given that \(\exp(\frac12\theta^2)\) is the moment generating function of the standard Normal random variable, estimate the least number of jumps such that there is a \(5\%\) chance that the frog lands 25~cm or more from his starting point.
A box contains \(n\) pieces of string, each of which has two ends. I select two string ends at random and tie them together. This creates either a ring (if the two ends are from the same string) or a longer piece of string. I repeat the process of tying together string ends chosen at random until there are none left. Find the expected number of rings created at the first step and hence obtain an expression for the expected number of rings created by the end of the process. Find also an expression for the variance of the number of rings created. Given that \(\ln 20 \approx 3\) and that \(1+ \frac12 + \cdots + \frac 1n \approx \ln n\) for large \(n\), determine approximately the expected number of rings created in the case \(n=40\,000\).