In Utopia there are three types of weather and on any particular day the weather belongs to just one of these: 1, sunny; 2, rainy; 3, cloudy but dry. It has been observed that if the weather on a certain day is of type \(i\) then that on the following day is of type \(j\) with probability \(p_{ij}\), where \(p_{ij}\) is the entry in the \(i\)th row and \(j\)th column of the array below:
Let \(X\) be a random variable which takes on only a finite number of different possible values, say \(x_1, x_2, \ldots, x_n\). Define the expectation of \(X\), \(E(X)\), and show that if \(a\) and \(b\) are constants then \(E(aX + b) = aE(X) + b\). Define also the variance of \(X\), \(\text{var}(X)\), and similarly express \(\text{var}(aX + b)\) in terms of \(\text{var}(X)\). By considering separately those \(x_i\) which satisfy \(|x_i - E(X)| > \epsilon\) and those which satisfy \(|x_i - E(X)| \leq \epsilon\) where \(\epsilon > 0\), show that $$P[|X - E(X)| > \epsilon] \leq \frac{\text{var}(X)}{\epsilon^2}.$$ If \(|x_i - E(X)| \leq \kappa\) for all \(i\), where \(\kappa > \epsilon\), show that $$P[|X - E(X)| > \epsilon] \geq \frac{\text{var}(X) - \epsilon^2}{\kappa^2 - \epsilon^2}.$$
For a certain mass-produced item the time that a randomly chosen individual lasts before failure may be supposed for practical purposes to be Normal with mean 100 and variance 1. A slight change is made in the conditions of manufacture, and the times until failure of \(n\) independently chosen items fail are determined, these being \(x_1, x_2, \ldots, x_n\). Construct a significance test at the 5\% level which would be appropriate in order to discover whether the mean length of life has increased, and explain carefully the meaning of such a procedure. (The variance may be supposed unchanged.) Determine how large \(n\) must be in order that the probability of not rejecting the null hypothesis is 0.05 if in fact the new mean is 101.
The integral $$I = \int_{x-h}^{x+h} f(u) du$$ is to be approximated by an expression of the form \(J = af(x-h) + bf(x) + cf(x+h)\), where \(a\), \(b\) and \(c\) may depend on \(h\) but are independent of the function \(f\) and of \(x\). Show that \(a\), \(b\) and \(c\) may be chosen in such a way that \(I = J\) whenever \(f\) is a polynomial of sufficiently low degree \(n\), and find the largest \(n\) for which this is true. Find values of \(a\), \(b\), \(c\) such that \(I = J\) whenever \(f(u) = p + q \sin u + r \cos u\).
A uniform ladder of length \(l\) and mass \(m\) stands on a smooth horizontal surface leaning against a smooth vertical wall. The foot of the ladder is subject to a force directed towards the wall of magnitude \(\lambda x^2\), where \(\lambda\) is a constant and \(x\) is the distance of the foot of the ladder from the wall. Find the condition that a man of mass \(M\) can stand in equilibrium at a distance \(y\) up the ladder. What happens if he slowly ascends the ladder?
The figure represents a vertical section through an ``overhead'' garage door. The door is rectangular, and of height 6 ft. On each side is a horizontal pin \(P\), 2 ft from the bottom edge, that slides smoothly in a vertical groove. The point \(Q\), 2 ft from the bottom edge, is connected by a light smoothly pivoted arm of length 2 ft to a joint \(R\) at the top of the groove 6 ft above the ground. From each pin \(P\) passes a vertical wire of weight taken over a pulley and carries at its other end a weight \(w\). The door is of weight \(W\), its centre of gravity \(G\) being equidistant from its top and bottom edges. Find the moment of the couple needed to hold the door in equilibrium at an angle \(\theta\) with the vertical, and show that, for a certain value of \(w/W\), \(L\) is zero for all positions of the door.
A gramophone record of mass \(m\) and radius \(a\) is placed on a horizontal turntable of radius greater than \(a\). The pressure between the record and the turntable is uniformly distributed and the coefficient of friction is \(\mu\). Show that, if, starting from zero, the angular acceleration \(f\) of the turntable is gradually increased, slipping takes place as soon as \(f\) exceeds \(4\mu g/3a\). A second record, of mass \(m'\) and radius \(a\), is placed upon the first, the coefficient of friction between the records being \(\mu'\). If, starting from zero, the angular acceleration of the turntable is gradually increased, find where slipping first takes place, and for what value of \(f\). If \(\mu = 3\mu'\), if the angular acceleration of the turntable is twice that at which slipping first occurs, and if \(I\) is the moment of inertia of the turntable, calculate the torque required to turn the turntable.
The displacement \(x\) of a simple harmonic oscillator satisfies the differential equation $$\frac{d^2x}{dt^2} + \omega^2 x = 0.$$ Denoting by \(\mathbf{M}\) and \(\mathbf{M}'\) the functions \(x = a \cos \omega(t - \alpha)\) and \(x = a' \cos \omega(t - \alpha')\), show that the functions \(\mathbf{M}\) and \(\mathbf{M} + \mathbf{M}'\), defined by $$x = \lambda a \cos \omega(t - \alpha) \text{ and } x = a \cos \omega(t - \alpha) + a' \cos \omega(t - \alpha')$$ respectively, satisfy the differential equation. Prove that the solutions of the equation form a vector space. What is its dimension? The scalar product \(\mathbf{M} \cdot \mathbf{M}'\) is defined as \(aa' \cos \omega(\alpha - \alpha')\). Prove that $$(\mathbf{M} + \mathbf{M}') \cdot \mathbf{M}'' = \mathbf{M} \cdot \mathbf{M}'' + \mathbf{M}' \cdot \mathbf{M}''.$$ Writing \(\mathbf{M} = m_1 \mathbf{e}_1 + m_2 \mathbf{e}_2\), where \(\mathbf{e}_1\) and \(\mathbf{e}_2\) denote the functions \(x = \cos \omega t\) and \(x = \sin \omega t\) respectively, express \(\mathbf{M} \cdot \mathbf{M}'\) in terms of the components \((m_1, m_2)\) and \((m_1', m_2')\). If the time \(t\) is measured from the instant \(t = 0\), so that \(t = t - 0\), and \(\mathbf{M}\) is written as \(\tilde{m}_1 \tilde{\mathbf{e}}_1 + \tilde{m}_2 \tilde{\mathbf{e}}_2\), where \(\tilde{\mathbf{e}}_1\) and \(\tilde{\mathbf{e}}_2\) are the functions \(x = \cos \omega t\) and \(x = \sin \omega t\) respectively, find the relation between the components \((\tilde{m}_1, \tilde{m}_2)\) and \((m_1, m_2)\). What is the nature of the matrix involved?
Two rings, each of mass \(m\), can slide along a rough horizontal rail; the coefficient of friction between the rings and the rail is \(\mu\). The rings are joined by a light inextensible inelastic string, to the mid-point of which is attached a particle of mass \(M\). The particle \(M\) is allowed to fall from rest, so that the two halves of the string become taut simultaneously: at this instant the angle between the two parts of the string is \(2\alpha\). The whole motion takes place in a vertical plane. Find the speed with which the rings are jerked into motion, given that the vertical velocity of the particle \(M\) just before the string becomes taut is \(V\).
Two spheres \(A\), \(B\) (not necessarily equal) are in direct collision, momentum being conserved. The velocities before the collision are \(u_A\), \(u_B\) and after the collision \(v_A\), \(v_B\), all velocities being measured in the direction from the centre of \(A\) to that of \(B\). Prove the equivalence of the following statements: