Show that, if \(X_1\), \(X_2\) and \(X_3\) are independent, and have a common continuous distribution, \(\text{Pr}[X_1 > \max(X_2, X_3)] = \frac{1}{3}\). Independent random samples \(X_1, X_2, ..., X_n\) and \(Y_1, Y_2, ..., Y_n\) are drawn from two unknown continuous distributions. It is suspected that, in fact, the distribution of the \(X\)'s is the same as that of the \(Y\)'s, and, to test this, it is proposed to consider the statistic \[W = \sum_{i,j=1}^{n} Z_{ij},\] where \[Z_{ij} = \begin{cases} 1 & \text{if} \quad X_i < Y_j \\ 0 & \text{if} \quad X_i \geq Y_j. \end{cases}\] Show that, if the two distributions are indeed identical, \(W\) has expectation \(n^2/2\) and variance \(n^2(2n + 1)/12\).* Show also that if, instead, \(P[X_i > Y_j] = p \neq \frac{1}{2}\), the expectation of \(W\) is \((1-p)n^2\). Assuming that \(W\) is approximately normally distributed, investigate whether the following data support the view that values from the \(Y\) distribution are typically larger than values from the \(X\) distribution: \(X\) \(3.41\) \(3.63\) \(3.77\) \(4.00\) \(4.54\) \(4.82\) \(4.91\) \(5.08\) \(Y\) \(3.91\) \(4.70\) \(4.71\) \(4.93\) \(4.95\) \(5.12\) \(5.37\) \(5.90\) * [Hint. Find the variance of \(W\) by first evaluating the expectation of \(W^2\).]
A particle of mass \(m\) moves in a horizontal straight line under a force equal to \(mn^2\) times the displacement from a point \(O\) in the line and directed towards \(O\). In addition, the motion of the particle is resisted by a force equal to \(mk\) times the square of its speed. The particle is projected with speed \(V\) from \(O\) along the line; by considering the motion in dimensionless form, show that the displacement \(x\) is of the form \[x = \frac{V}{n}f\left(\frac{v}{V}, \frac{kV}{n}\right),\] where \(v\) is the non-zero speed of the particle when at \(x\). Show that the particle first comes to rest at distance \(X\), of the form \[X = \frac{V}{n}g\left(\frac{kV}{n}\right).\] If \(\frac{kV}{n}\) is small, show that \(g\left(\frac{kV}{n}\right)\) is approximately \[1-\frac{2kV}{3n}.\]
A particle moves in a horizontal circle on the inner surface of a smooth spherical shell of radius \(a\) and is slightly disturbed. Show that the period of the small oscillation about the steady motion is \[2\pi \left(\frac{a \cos \alpha}{g(1+3\cos^2 \alpha)}\right)^{\frac{1}{2}},\] where \(a \sin \alpha\) is the radius of the circle.
A particle of unit mass is projected from level ground with speed \(u\sqrt{2}\) at an elevation of \(\frac{1}{4}\pi\) above the horizontal. It experiences a resisting force directly opposing its motion whose magnitude is \(k\) times the square of the particle's speed, where \(ku^2/g\) is a small number. Show that, after a lapse of time \(t\) during the flight, the vertical component of the acceleration is approximately equal to \[-g-k(u-gt) \{(u-gt)^2 + u^2t^2\}^{\frac{1}{2}}.\] Deduce that the highest point of the trajectory is attained when \(t\) is approximately equal to \[\frac{u}{g} - \frac{ku^3}{3g^2}(2^{\frac{1}{2}}-1).\]
The moment of relative momentum of a particle \(P\), of mass \(m\), about an arbitrary point \(O'\) is defined as \(m\mathbf{r'} \wedge \dot{\mathbf{r'}}\), where \(\mathbf{r'} = \overrightarrow{O'P}\) and the dot denotes differentiation with respect to time. A collection of particles has centre of gravity \(G\), total mass \(M\), and moment of momentum \(\mathbf{h}\) about the origin \(O\) of a fixed coordinate system. Show that the moment of relative momentum about an arbitrary point \(O'\) is \(\mathbf{h'}\), given by \[\mathbf{h'} = \mathbf{h} - M\mathbf{s} \wedge \dot{\mathbf{f}} - M(\mathbf{f}-\mathbf{s}) \wedge \dot{\mathbf{s}},\] where \(\mathbf{f} = \overrightarrow{OG}\) and \(\mathbf{s} = \overrightarrow{OO'}\). Show also that, if \(\mathbf{L'}\) is the moment of the external forces about \(O'\), then \[\dot{\mathbf{h'}} = \mathbf{L'} - M(\mathbf{f}-\mathbf{s}) \wedge \ddot{\mathbf{s}}.\]
Show that for three vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) \[(\mathbf{a} \wedge \mathbf{b})\cdot(\mathbf{a} \wedge \mathbf{c}) = (\mathbf{a}\cdot\mathbf{a})(\mathbf{b}\cdot\mathbf{c})-(\mathbf{a}\cdot\mathbf{b})(\mathbf{a}\cdot\mathbf{c})\] and \[(\mathbf{a} \wedge \mathbf{b}) \wedge (\mathbf{a} \wedge \mathbf{c}) = \mathbf{a}(\mathbf{a}\cdot\mathbf{b} \wedge \mathbf{c}).\] [You may assume \[\mathbf{x} \wedge \mathbf{y}\cdot\mathbf{z} = \mathbf{x} \cdot \mathbf{y} \wedge \mathbf{z}\] and \[(\mathbf{x} \wedge \mathbf{y}) \wedge \mathbf{z} = \mathbf{y}(\mathbf{x}\cdot\mathbf{z}) - \mathbf{x}(\mathbf{y}\cdot\mathbf{z}).\] Three points \(A\), \(B\) and \(C\) lie on a sphere with centre \(O\). Let \(\hat{A}\), \(\hat{B}\) and \(\hat{C}\) be the angles \(BOC\), \(COA\) and \(AOB\), and let \(\alpha\), \(\beta\) and \(\gamma\) be the angles between the pairs of planes \(AOB\) \& \(AOC\), \(BOC\) \& \(BOA\) and \(COA\) \& \(COB\). Deduce the spherical triangle cosine and sine formulae \[\cos \hat{A} = \cos \hat{B} \cos \hat{C} + \sin \hat{B} \sin \hat{C} \cos \alpha\] and \[\frac{\sin \alpha}{\sin \hat{A}} = \frac{\sin \beta}{\sin \hat{B}} = \frac{\sin \gamma}{\sin \hat{C}}.\]
Suppose that \(a\), \(b\) and \(c\) are real numbers such that the equation \[x^3-ax^2+bx-c=0\] has three distinct real roots, which are in geometric progression. Prove that \(abc > 0\) and that \[\left|\frac{a}{c}-1\right| > 2.\]
Express \((a^2+b^2+c^2)(x^2+\beta^2+\gamma^2)-(a\alpha+b\beta+c\gamma)^2\) as the sum of three squares. Deduce that if \(\alpha\), \(\beta\), \(\gamma\) are real numbers then \[(a^4+\beta^4+\gamma^4)(a^2+\beta^2+\gamma^2) \geq (a^2+\beta^2+\gamma^2)^2.\] Give necessary and sufficient conditions on \(\alpha\), \(\beta\), \(\gamma\) for equality to hold.
Suppose that \(n\), \(x\) and \(y\) are positive integers such that \(n+x\) is a square and \(n+y\) is the next larger square. Show that \(n+xy\) and \(n+xy+x+y\) are adjacent squares. Hence show that \(n+x+y+xy(2+x+y+xy)\) is a square.
Solution: \begin{align*} && n+x &= k^2\\ && n+y &= (k+1)^2 = n+x+2k+1 \\ \Rightarrow && y &= x + 2k+1 \\ && n +xy &= n + x(x+2k+1) \\ &&&= n+x^2+2kx+x \\ &&&= (k^2-x)+x^2+2kx+x \\ &&&= k^2+x^2+2kx \\ &&&=(k+x)^2 \\ &&n+xy+x+y &= k^2-x+x(x+2k+1)+x+x+2k+1 \\ &&&= k^2-x+x^2+2kx+x+2x+2k+1 \\ &&&= k^2+x^2+2kx+2x+2k+1 \\ &&&=(k+x+1)^2 \\ \\ && n+x+y+xy(2+x+y+xy) &= (k+x)^2+x+y+xy+xy(x+y+xy) \\ &&&= (k+x)^2+(x+y+xy)(1+xy) \\ &&&= (k+x)^2+((k+x+1)^2-n)(1+(k+x)^2-n) \\ &&&= (k+x)^2+((k+x)^2+2(k+x)+1-n)((k+x)^2+1-n) \\ &&&= (k+x)^2+(k+x)^4+(1-n)^2+(k+x)^2(2(k+x)+2(1-n))+2(k+x)(1-n) \\ &&&= (k+x)^4+(k+x)^2( \end{align*}
Show that the number of ways of arranging \(N\) indistinguishable oranges and \(M\) indistinguishable pencils in a line is \[\frac{(N+M)!}{N!M!}.\] Hence or otherwise calculate the number of ways of putting \(N\) indistinguishable oranges into \(P\) boxes numbered \(1, \ldots, P\).