Part IA, 2001, Paper 2

# Part IA, 2001, Paper 2

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2.I.1B

commentFind the solution to

$\frac{d y(x)}{d x}+\tanh (x) y(x)=H(x)$

in the range $-\infty<x<\infty$ subject to $y(0)=1$, where $H(x)$ is the Heavyside function defined by

$H(x)= \begin{cases}0 & x<0 \\ 1 & x>0\end{cases}$

Sketch the solution.

2.I.2B

commentThe function $y(x)$ satisfies the inhomogeneous second-order linear differential equation

$y^{\prime \prime}-y^{\prime}-2 y=18 x e^{-x} .$

Find the solution that satisfies the conditions that $y(0)=1$ and $y(x)$ is bounded as $x \rightarrow \infty$.

2.II.5B

commentThe real sequence $y_{k}, k=1,2, \ldots$ satisfies the difference equation

$y_{k+2}-y_{k+1}+y_{k}=0$

Show that the general solution can be written

$y_{k}=a \cos \frac{\pi k}{3}+b \sin \frac{\pi k}{3},$

where $a$ and $b$ are arbitrary real constants.

Now let $y_{k}$ satisfy

$y_{k+2}-y_{k+1}+y_{k}=\frac{1}{k+2}$

Show that a particular solution of $(*)$ can be written in the form

$y_{k}=\sum_{n=1}^{k} \frac{a_{n}}{k-n+1},$

where

$a_{n+2}-a_{n+1}+a_{n}=0, \quad n \geq 1$

and $a_{1}=1, a_{2}=1$.

Hence, find the general solution to $(*)$.

2.II.7B

commentConsider the linear system

$\dot{\mathbf{z}}+A \mathbf{z}=\mathbf{h}$

where

$\mathbf{z}(t)=\left(\begin{array}{c} x(t) \\ y(t) \end{array}\right), \quad A=\left(\begin{array}{cc} 1+a & -2 \\ 1 & -1+a \end{array}\right), \quad \mathbf{h}(t)=\left(\begin{array}{c} 2 \cos t \\ \cos t-\sin t \end{array}\right)$

where $\mathbf{z}(t)$ is real and $a$ is a real constant, $a \geq 0$.

Find a (complex) eigenvector, e, of $A$ and its corresponding (complex) eigenvalue, $l \underline{l}$. Show that the second eigenvector and corresponding eigenvalue are respectively $\overline{\mathbf{e}}$ and $\bar{l}$, where the bar over the symbols signifies complex conjugation. Hence explain how the general solution to $(*)$ can be written as

$\mathbf{z}(t)=\alpha(t) \mathbf{e}+\bar{\alpha}(t) \overline{\mathbf{e}},$

where $\alpha(t)$ is complex.

Write down a differential equation for $\alpha(t)$ and hence, for $a>0$, deduce the solution to $(*)$ which satisfies the initial condition $\mathbf{z}(0)=\underline{0}$.

Is the linear system resonant?

By taking the limit $a \rightarrow 0$ of the solution already found deduce the solution satisfying $\mathbf{z}(0)=0$ when $a=0$.

2.II.8B

commentCarnivorous hunters of population $h$ prey on vegetarians of population $p$. In the absence of hunters the prey will increase in number until their population is limited by the availability of food. In the absence of prey the hunters will eventually die out. The equations governing the evolution of the populations are

$\begin{aligned} \dot{p} &=p\left(1-\frac{p}{a}\right)-\frac{p h}{a}, \\ \dot{h} &=\frac{h}{8}\left(\frac{p}{b}-1\right), \end{aligned}$

where $a$ and $b$ are positive constants, and $h(t)$ and $p(t)$ are non-negative functions of time, $t$. By giving an interpretation of each term explain briefly how these equations model the system described.

Consider these equations for $a=1$. In the two cases $0<b<1 / 2$ and $b>1$ determine the location and the stability properties of the critical points of $(*)$. In both of these cases sketch the typical solution trajectories and briefly describe the ultimate fate of hunters and prey.

2.I.3F

commentThe following problem is known as Bertrand's paradox. A chord has been chosen at random in a circle of radius $r$. Find the probability that it is longer than the side of the equilateral triangle inscribed in the circle. Consider three different cases:

a) the middle point of the chord is distributed uniformly inside the circle,

b) the two endpoints of the chord are independent and uniformly distributed over the circumference,

c) the distance between the middle point of the chord and the centre of the circle is uniformly distributed over the interval $[0, r]$.

[Hint: drawing diagrams may help considerably.]

2.I.4F

commentThe Ruritanian authorities decided to pardon and release one out of three remaining inmates, $A, B$ and $C$, kept in strict isolation in the notorious Alkazaf prison. The inmates know this, but can't guess who among them is the lucky one; the waiting is agonising. A sympathetic, but corrupted, prison guard approaches $A$ and offers to name, in exchange for a fee, another inmate (not $A)$ who is doomed to stay. He says: "This reduces your chances to remain here from $2 / 3$ to $1 / 2$ : will it make you feel better?" $A$ hesitates but then accepts the offer; the guard names $B$.

Assume that indeed $B$ will not be released. Determine the conditional probability

$P(A \text { remains } \mid B \text { named })=\frac{P(A \& B \text { remain })}{P(B \text { named })}$

and thus check the guard's claim, in three cases:

a) when the guard is completely unbiased (i.e., names any of $B$ and $C$ with probability $1 / 2$ if the pair $B, C$ is to remain jailed),

b) if he hates $B$ and would certainly name him if $B$ is to remain jailed,

c) if he hates $C$ and would certainly name him if $C$ is to remain jailed.

2.II.10F

commentA random point is distributed uniformly in a unit circle $\mathcal{D}$ so that the probability that it falls within a subset $\mathcal{A} \subseteq \mathcal{D}$ is proportional to the area of $\mathcal{A}$. Let $R$ denote the distance between the point and the centre of the circle. Find the distribution function $F_{R}(x)=P(R<x)$, the expected value $E R$ and the variance $\operatorname{Var} R=E R^{2}-(E R)^{2}$.

Let $\Theta$ be the angle formed by the radius through the random point and the horizontal line. Prove that $R$ and $\Theta$ are independent random variables.

Consider a coordinate system where the origin is placed at the centre of $\mathcal{D}$. Let $X$ and $Y$ denote the horizontal and vertical coordinates of the random point. Find the covariance $\operatorname{Cov}(X, Y)=E(X Y)-E X E Y$ and determine whether $X$ and $Y$ are independent.

Calculate the sum of expected values $E \frac{X}{R}+i E \frac{Y}{R}$. Show that it can be written as the expected value $E e^{i \xi}$ and determine the random variable $\xi$.

2.II.11F

commentDipkomsky, a desperado in the wild West, is surrounded by an enemy gang and fighting tooth and nail for his survival. He has $m$ guns, $m>1$, pointing in different directions and tries to use them in succession to give an impression that there are several defenders. When he turns to a subsequent gun and discovers that the gun is loaded he fires it with probability $1 / 2$ and moves to the next one. Otherwise, i.e. when the gun is unloaded, he loads it with probability $3 / 4$ or simply moves to the next gun with complementary probability $1 / 4$. If he decides to load the gun he then fires it or not with probability $1 / 2$ and after that moves to the next gun anyway.

Initially, each gun had been loaded independently with probability $p$. Show that if after each move this distribution is preserved, then $p=3 / 7$. Calculate the expected value $E N$ and variance Var $N$ of the number $N$ of loaded guns under this distribution.

[Hint: it may be helpful to represent $N$ as a sum $\sum_{1 \leq j \leq m} X_{j}$ of random variables taking values 0 and 1.]

2.II.12F

commentA taxi travels between four villages, $W, X, Y, Z$, situated at the corners of a rectangle. The four roads connecting the villages follow the sides of the rectangle; the distance from $W$ to $X$ and $Y$ to $Z$ is 5 miles and from $W$ to $Z$ and $Y$ to $X 10$ miles. After delivering a customer the taxi waits until the next call then goes to pick up the new customer and takes him to his destination. The calls may come from any of the villages with probability $1 / 4$ and each customer goes to any other village with probability $1 / 3$. Naturally, when travelling between a pair of adjacent corners of the rectangle, the taxi takes the straight route, otherwise (when it travels from $W$ to $Y$ or $X$ to $Z$ or vice versa) it does not matter. Distances within a given village are negligible. Let $D$ be the distance travelled to pick up and deliver a single customer. Find the probabilitites that $D$ takes each of its possible values. Find the expected value $E D$ and the variance Var $D$.