In a certain chemical reaction 1 mole of a product \(P\) is produced per mole of reactant \(R\). The rate of production of \(P\) in moles per litre per second is \(k\) times the product of the concentrations of \(P\) and \(R\), these concentrations being measured in moles per litre. Initially there is 1 mole of \(P\) present for every 100 moles of \(R\). Assuming that the system is closed and has constant volume, i.e., that the sum of the concentrations of \(P\) and \(R\) is some constant \(\alpha\), calculate, in terms of \(\alpha\) and \(k\), the time that elapses before there are 100 moles of \(P\) present for every mole of \(R\).
Suppose that the random variable \(X\) has cumulative distribution function \(F(x)\) (which is the probability that \(X\) is less than or equal to \(x\)) and probability density function \(f(x)\) (which is \(F'(x)\)). If \(F(x) = 1-e^{-\lambda x}\) for \(x \geq 0\), and \(F(x) = 0\) for \(x < 0\), find the mean and variance of \(X\). A business man awaits an order from each of \(n\) clients. The \(n\) orders are sent out simultaneously, and the times taken to reach the business man are independent random variables, each with density function \(\lambda e^{-\lambda x}\) (\(x \geq 0\)). The first order to be received will be dispatched free of charge to the client. How long should the business man expect to wait before dispatching the free order? [Hint: the minimum of \(n\) variables, \(X_1, ..., X_n\) is greater than \(x\) if and only if each of \(X_1, ..., X_n\) is greater than \(x\).] What chance has a particular client of getting his order free?
A random sample \(X_1 ... X_n\) is taken from the normal distribution with mean \(\mu\) and variance \(\sigma^2\). Find the mean and variance of \(\overline{X}\), the sample mean, and find the expected value of \(\sum_{i=1}^{n} (X_i-\overline{X})^2\). For what value of \(k\) will \(k \sum_{i=1}^{n} (X_i-\overline{X})^2\) be an unbiased estimator of \(\sigma^2\), i.e. have expected value \(\sigma^2\)? In an experiment to determine the growth rate of human infants, nine randomly selected infants are fed with an approved diet for two weeks, and their weight gains \(X_1,..., X_9\) during that period are recorded in pounds. It is observed that \(\overline{X} = 1.2\) and \(\sum_{i=1}^{9}(X_i-\overline{X})^2 = 0.72\). Use the tables of the \(t\)-distribution to find a 95\% confidence interval for the mean weight gain. Medical science dictates that the approved growth rate is an ounce a day. Do these babies conform to the approved rate?
A method for the hospital diagnosis of the presence or absence of a minor illness costs the hospital £\(C\) to apply. The probability of wrongly diagnosing a patient as 'well' is \(\alpha\), and the probability of wrongly diagnosing him as 'ill' is \(\beta\). If a patient is wrongly diagnosed as 'well', the cost to the hospital is assessed as £\(K\); if he is wrongly diagnosed as 'ill', the cost is assessed as £\(K'\). Correct diagnoses incur no further cost. The incidence of the illness is thought to be 1 in every 100 of the population. Find the expected cost of diagnosing a patient with this method. [You may assume this expected cost is [(expected total cost of diagnosing a patient who is ill) \(\times\) 0.01 + (expected total cost of diagnosing a patient who is well) \(\times\) 0.99].] Two methods I and II are available with costs \(C_1\), \(C_2\) respectively, and error probabilities \((\alpha_1, \beta_1)\), \((\alpha_2, \beta_2)\) respectively (with \(C\)'s, \(\alpha\)'s and \(\beta\)'s defined as above). Find which method has the smaller expected cost if \(C_1 = 1\), \(C_2 = 2\), \(\alpha_1 = 0.4\), \(\beta_1 = 0.05\), \(\alpha_2 = 0.3\), \(\beta_2 = 0.1\).
An anthropologist encounters a large group of savages in the jungle. He knows that either they all come from tribe \(A\) or they all come from tribe \(B\). In both cases their heights are independently distributed; if they are from \(A\) then the heights are normal with mean \(\mu_A = 60\) inches and standard deviation \(\sigma = 5\) inches; if they are from \(B\) the heights are normal with mean \(\mu_B = 66\) inches, and standard deviation \(\sigma = 5\) inches. In order to decide to which tribe they belong, the anthropologist uses a rule of the following form. He assigns them to \(A\) if \(\overline{x}_n < \xi\), and otherwise to \(B\), where \(\overline{x}_n\) is the mean of the heights of \(n\) savages. Show how he should choose \(\xi\) in order that \(\alpha\), the probability of wrongly assigning them to \(B\), is 0.05. Find the corresponding value of \(\beta\), the probability of wrongly assigning them to \(A\), and find how large \(n\) should be in order that \(\beta\) is 0.01 or less. [You may assume that \(\overline{x}_n\) has a normal distribution, whose mean depends on whether the savages are from \(A\) or \(B\).]
Assume that for all \(x\) such that \(|x| < 1\), \(\sin^{-1}x = \sum_{r=0}^{\infty} \frac{(2r)!}{2^{2r}(r!)^2}\frac{x^{2r+1}}{2r+1}\). Writing \(u_r\) for the coefficient of \(x^{2r+1}\) in the above expansion, show that \(\frac{u_r}{u_{r-1}} = \frac{(2r-1)^2}{2r(2r+1)} < 1\), for all \(r \geq 1\). By quoting this series with \(x = \frac{1}{2}\), express \(\pi\) as the sum of a series of positive terms; hence construct a flow diagram to calculate \(\pi\), accumulating terms up to and including the first whose value is less than \(10^{-10}\). Prove that the value of \(\pi\) computed is correct to within \(\frac{1}{3} \cdot 10^{-10}\).
A uniform solid, with total mass \(M\), occupies the volume obtained by rotating about the \(x\)-axis the area lying between the two parabolas \(y^2 = 4ax\) \((0 < x < b)\) and \(y^2 = 8ax - 4ab\) \((\frac{1}{2}b < x < b)\). Find the position of its mass centre and calculate its moment of inertia about the \(x\)-axis.
The surface of a lawn is a plane inclined to the horizontal at an angle \(\alpha\). A sprinkler is embedded in the surface, and emits droplets of water in all directions, the speed of projection being \(v\). Show that the region watered is an ellipse with area \(\pi v^4/g^2\cos^2\alpha\).
A circular disc rolls without slipping along a straight line, with uniform angular velocity. Show that the acceleration of each point of the disc is directed towards the centre. Discuss, without making detailed calculations, whether the same result holds if the disc rolls with non-uniform angular velocity.
Two equal uniform rods \(AB, BC\), each of length \(2a\) and weight \(W\), are freely jointed at \(B\). The angle \(ABC\) is maintained at a value \(2\alpha\) by means of a light string \(AC\). The rods are in equilibrium in a vertical plane with \(AB\) and \(BC\) resting on two small smooth pegs \(P, Q\), where \(PQ\) is horizontal and of length \(2c\) \((c > a\sin^3\alpha)\), and \(B\) is vertically above the midpoint of \(AC\). Show that the tension in \(AC\) is \(\frac{W\tan\alpha(c\textrm{cosec}^3\alpha-a)}{2a}\).