Problems

Defining the coefficient of correlation between two variables \(x\) and \(y\) as \(\rho = \frac{E[(x-Ex)(y-Ey)]}{\sqrt{E[(x-Ex)^2]} \cdot \sqrt{E[(y-Ey)^2]}},\) where \(E\) denotes the expectation value, prove

1. [(i)] \(|\rho| \leq 1\),
2. [(ii)] \(\rho = 1\) if \(y = ax + b\), where \(a\) and \(b\) are constants and \(a > 0\),
3. [(iii)] if \(X = px + q\) and \(Y = ry + s\), where \(p\), \(q\), \(r\) and \(s\) are constants and \(p\) and \(r\) are positive, the coefficient of correlation between \(X\) and \(Y\) is equal to that between \(x\) and \(y\).

There are 50,000 shares in a lottery with 1000 prizes. If a syndicate buys 100 shares, write down an approximate expression for, and evaluate approximately, the chance that it wins four or more prizes. Find also the variance of the number of prizes that may be expected. [The approximation \((1 + 1/n)^{xn} = e^x\) for large \(n\) may be used.]

If \(\alpha\) is a complex fifth root of unity, prove that \(\alpha - \alpha^4\) is a root of the equation $$\alpha^4 + 5\alpha^2 + 5 = 0.$$ Express the other roots of this equation in terms of \(\alpha\).

If \(u = x + y\), \(v = xy\), and \(x^n + y^n = 1\), find the degree in \(v\) of the algebraic relation between \(u\) and \(v\). If \(n = 5\), prove that $$5(x + y)(1 - x)(1 - y)(1 - x - y + x^2 + xy + y^2) = (x + y - 1)^5.$$

Prove that, if \(x_1\) and \(x_2\) are connected by the relation $$ax_1x_2 + bx_1 + cx_2 + d = 0,$$ there are, in general, two unequal values \(m\), \(n\) of \(x_1\) for which \(x_2 = x_1\); and that the relation is equivalent to $$\frac{x_2 - m}{x_2 - n} = k \frac{x_1 - m}{x_1 - n},$$ where \(k\) is a root of the equation $$(bc - ad)(k^2 + 1) + (b^2 + c^2 - 2ad)k = 0.$$ Find the relation between \(a\), \(b\), \(c\), \(d\) in order that the equations \begin{align} ax_1x_2 + bx_1 + cx_2 + d &= 0,\\ ax_1x_3 + bx_2 + cx_3 + d &= 0,\\ ax_2x_4 + bx_3 + cx_4 + d &= 0,\\ ax_3x_4 + bx_4 + cx_1 + d &= 0 \end{align} may be satisfied by values of \(x_1\), \(x_2\), \(x_3\), \(x_4\) which are all different.

Write down the (complex) factors of \(x^2 + y^2 + z^2 - yz - zx - xy\). If \(x\), \(y\), \(z\), \(a\), \(b\), \(c\) are real and \(ax + by + cz = 0\), prove that the product of $$\frac{x^2 + y^2 + z^2 - yz - zx - xy}{(x + y + z)^3} \text{ and } \frac{a^3 + b^3 + c^3 - bc - ca - ab}{(a + b + c)^3}$$ cannot be less than \(\frac{1}{4}\). Find the ratios \(x:y:z\) in terms of \(a\), \(b\), \(c\) if the product is equal to \(\frac{1}{4}\).

Express \(\tan n\theta\) in terms of \(\tan \theta\), where \(n\) is a positive integer. If \(n\) is odd, prove that $$n \tan n\theta = 1 + 2 \sum_{r=1}^{(n-1)/2} \frac{\sec^2(2r-1)\alpha}{\tan^3(2r-1)\alpha - \tan^3 \theta},$$ where \(2n\alpha = \pi\).

A flagstaff leaning due north at an angle \(\alpha\) to the vertical subtends angles \(\phi_1\) and \(\phi_2\) respectively, from two points \(P_1\) and \(P_2\) on a horizontal road leading north-west from its base. Prove that the length of the flagstaff is $$\frac{\pm \sqrt{2b} \sin \phi_1 \sin \phi_2}{\sin(\phi_1 - \phi_2)(2 - \sin^2 \alpha)^{1/2}},$$ where \(b\) is the distance \(P_1P_2\).

Prove that the area of the greatest equilateral triangle which can be drawn with its three sides passing respectively through three given points \(A\), \(B\), \(C\) is $$2\Delta + \frac{a^2 + b^2 + c^2}{2\sqrt{3}},$$ where \(a\), \(b\), \(c\) are the sides of the triangle \(ABC\), and \(\Delta\) is its area.

If \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\), prove that $$y_3 = \frac{d^2y}{dx^2} = \frac{k}{(hx + by + f)^3},$$ where \(k\) is a constant. Prove also that $$\frac{d}{dx}\left(\frac{1}{y_3}\frac{d^2}{dx^2}(y_3^{-1})\right) = 0,$$ and express this result rationally in terms of derivatives of \(y\) with respect to \(x\).