The president of the republic must have a son and heir. It may be assumed that each baby born to him is equally likely to be a boy or a girl, irrespective of the sexes of his previous children. Let \(\mu\) and \(\sigma^2\) denote the mean and variance, respectively, of the number of children in his family, if he decides to have no more children once he has a son. Evaluate \(\mu\). Now suppose that he decides to have no more children once he has exactly \(r\) sons. Express the mean and variance of \(C\), the number of children in his family, in terms of \(\mu\) and \(\sigma^2\). By considering the numbers of boys and girls among \(2r - 1\) children, or otherwise, show that \(\Pr[C < 2r] = \frac{1}{2}\).
Let \(X_1, X_2, \ldots, X_n\) be independent and identically distributed random variables with mean \(\beta\), taking integer values in the range \(1, 2, \ldots, K\). For each \(m\), \(1 \leq m \leq n\), let \(S_m = X_1 + X_2 + \cdots + X_m\). Prove that \(E(X_r/S_m) = 1/m\) for \(r = 1, 2, \ldots, m\). Hence show that, if \(m \leq n\), \(E(S_m/S_n) = m/n\) and \(E(S_n/S_m) = 1 + (n-m)\beta E(1/S_m)\).
The average weight in grams of the contents of a sachet of instant mashed potato varies between batches, because of the variable quality of the synthetic feedstock. Within a given batch, the weights of the sachets are independently and normally distributed, with common unknown mean \(m\) and standard deviation \(0 \cdot 1\). In order to check the weight of a given batch, the manufacturer weighs the contents of 25 sachets, obtaining an average weight of \(4 \cdot 92\). Does this give him good grounds for rejecting the hypothesis that \(m\) is really 5? He now decides upon the policy of rejecting a batch whenever the average weight of a sample of \(N\) sachets falls below \(T\). If \(N\) and \(T\) are to be chosen so that the probabilities of wrongly rejecting a batch with \(m = 5\) and of wrongly accepting a batch with \(m = 4 \cdot 95\) are both less than \(0 \cdot 05\), what values would you choose to make \(N\) as small as possible?
A machine produces boiled sweets in large batches. Each batch is either satisfactory, and contains no sub-standard sweets, or defective, when a known proportion \(p\) of the sweets are tasteless. The cost of rejecting a defective batch immediately after production is \(K\); however, if a defective batch is not detected and reaches the customer, it costs \(MK\) to replace it and to recover lost goodwill, where \(M > 1\). The quality control officer decides to test each batch by removing, for tasting, a random number \(N\) of sweets selected from it at random, at a cost of \(c\) per sweet, where \(N\) has a Poisson distribution with mean \(\lambda\). A batch is rejected if any of the \(N\) sweets proves to be tasteless. Show that his chance of detecting a defective batch is \(1 - e^{-\lambda p}\). If the proportion of defective batches produced is known to be \(\alpha\), show that the expected running cost per batch is \(c\lambda + \alpha K[1 + (M-1)e^{-\lambda p}]\). Find the value of \(\lambda\) that minimizes the expected cost (a) if \(c < \alpha p K(M-1)\), and (b) if \(c \geq \alpha p K(M-1)\).
Prove that \[\int_0^{2\pi} \sin nx \sin mx\, dx = 0\] when the positive integers \(n\) and \(m\) are not equal, and evaluate the integral for the case when \(n = m\). Let \(f(x)\) be a periodic function with period \(2\pi\), which may be expressed in the form \[f(x) = \sum_{n=1}^{\infty} a_n \sin nx\] for some constants \(a_n\). Use the results of the first part to obtain expressions for the \(a_n\) in terms of \(f(x)\) by multiplying by \(\sin mx\) and integrating term by term. [You may assume that this procedure is justified.] We now seek, for fixed \(N\), to choose \(a_n\) so that the sum \[\sum_{n=1}^{N} a_n \sin nx\] approximates \(f(x)\) as closely as possible, in the sense that \[\int_0^{2\pi} \left\{f(x) - \sum_{n=1}^{N} a_n \sin nx\right\}^2\, dx\] is minimal. By differentiating with respect to each \(a_n\) separately, show that the solution is given by \(a_n = \hat{a}_n\).
An arthritic squash player cannot move from the point where he is placed initially, and can project the ball only with a fixed velocity in a fixed direction. Since no-one will play with him he bounces the ball back exactly to himself, with a single bounce off a wall, not the floor or ceiling). If the coefficient of restitution at the bounce is \(e\), show that the distance from the wall at which he should have himself positioned is proportional to \(e/(1+e)\).
A railway truck of total mass \(M\) has identical wheels of radius \(a\) whose combined moment of inertia, about the axles is \(J\). The axle bearings are frictionless, but the coefficient of limiting friction between the wheels and the rails is \(\mu\). The truck is on a horizontal track, and is pulled by a force \(P\). The vertical acceleration due to gravity is \(g\).
A weightless rod carries a particle of mass \(m\) at its upper end. It is balanced in unstable equilibrium on a rough horizontal table, and begins to fall sideways. Using conservation of energy, find the angular velocity (squared) and the angular acceleration as functions of the angle \(\theta\) through which it has fallen, assuming the lower end does not move. Use these to show that the vertical component of force, where the rod touches the table, is \[N = mg(3\cos^2\theta - 2\cos\theta),\] and find the horizontal component. Let the coefficient of friction between the rod and the table be \(\mu\). Show that the rod's lower end either leaves the surface of the table when \(\cos\theta = \frac{1}{3}\), or slips when \(\tan\theta = \mu\). What determines which happens?
A smooth ring of elastic material (modulus of elasticity \(\lambda\)) has natural radius \(R\), negligible cross section, and mass \(M\). A smooth-sided right circular cone, whose vertex angle is \(2\alpha\), is held fixed with its axis vertical and vertex uppermost. The ring is placed over the cone so that it is always in contact with the cone, and moves so that the plane of the ring is always horizontal. If \(x\) is the distance between the centre of the ring and the vertex of the cone, show that the potential energy of the ring is, to within an additive constant, \begin{align*} \frac{\pi\lambda}{R}(x\tan\alpha - R)^2 - Mgx, \end{align*} where \(g\) is the acceleration due to gravity. By considering the total energy of the ring (or otherwise), find the equilibrium position. Show that when disturbed the ring oscillates about this position with simple harmonic motion, and find the period. [Modulus of elasticity \(\lambda\) is defined so that tension=\(\lambda \times\) (extension)\(\div\)(natural length).]
A particle of mass \(m\) is projected with velocity \(U\) horizontally and \(V\) vertically; gravity is constant with magnitude \(g\). Obtain the components of velocity as functions of time, and find the time of flight and the range to the point where it returns to its starting level. Slight air resistance, providing a force \(mk\) times the velocity, has now to be allowed for; \(k V/g\) is much less than unity. Approximating the vertical resistive force by using the velocity component found in (i), or otherwise, show that to first order in \(k V/g\) the time of flight is decreased by a fraction \(kV/3g\).