Part IA, 2015, Paper 2

# Part IA, 2015, Paper 2

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Paper 2, Section $I$, B

commentFind the general solution of the equation

$2 \frac{d y}{d t}=y-y^{3} .$

Compute all possible limiting values of $y$ as $t \rightarrow \infty$.

Find a non-zero value of $y(0)$ such that $y(t)=y(0)$ for all $t$.

Paper 2, Section I, B

commentFind the general solution of the equation

$\frac{d y}{d x}-2 y=e^{\lambda x}$

where $\lambda$ is a constant not equal to 2 .

By subtracting from the particular integral an appropriate multiple of the complementary function, obtain the limit as $\lambda \rightarrow 2$ of the general solution of $(*)$ and confirm that it yields the general solution for $\lambda=2$.

Solve equation $(*)$ with $\lambda=2$ and $y(1)=2$.

Paper 2, Section II, B

commentConsider the equation

$2 \frac{\partial^{2} u}{\partial x^{2}}+3 \frac{\partial^{2} u}{\partial y^{2}}-7 \frac{\partial^{2} u}{\partial x \partial y}=0$

for the function $u(x, y)$, where $x$ and $y$ are real variables. By using the change of variables

$\xi=x+\alpha y, \quad \eta=\beta x+y$

where $\alpha$ and $\beta$ are appropriately chosen integers, transform $(*)$ into the equation

$\frac{\partial^{2} u}{\partial \xi \partial \eta}=0$

Hence, solve equation $(*)$ supplemented with the boundary conditions

$u(0, y)=4 y^{2}, \quad u(-2 y, y)=0, \quad \text { for all } y$

Paper 2, Section II, B

commentWrite as a system of two first-order equations the second-order equation

$\frac{d^{2} \theta}{d t^{2}}+c \frac{d \theta}{d t}\left|\frac{d \theta}{d t}\right|+\sin \theta=0$

where $c$ is a small, positive constant, and find its equilibrium points. What is the nature of these points?

Draw the trajectories in the $(\theta, \omega)$ plane, where $\omega=d \theta / d t$, in the neighbourhood of two typical equilibrium points.

By considering the cases of $\omega>0$ and $\omega<0$ separately, find explicit expressions for $\omega^{2}$ as a function of $\theta$. Discuss how the second term in $(*)$ affects the nature of the equilibrium points.

Paper 2, Section II, B

commentSuppose that $\mathbf{x}(t) \in \mathbb{R}^{3}$ obeys the differential equation

$\frac{d \mathbf{x}}{d t}=M \mathbf{x}$

where $M$ is a constant $3 \times 3$ real matrix.

(i) Suppose that $M$ has distinct eigenvalues $\lambda_{1}, \lambda_{2}, \lambda_{3}$ with corresponding eigenvectors $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$. Explain why $\mathbf{x}$ may be expressed in the form $\sum_{i=1}^{3} a_{i}(t) \mathbf{e}_{i}$ and deduce by substitution that the general solution of $(*)$ is

$\mathbf{x}=\sum_{i=1}^{3} A_{i} e^{\lambda_{i} t} \mathbf{e}_{i}$

where $A_{1}, A_{2}, A_{3}$ are constants.

(ii) What is the general solution of $(*)$ if $\lambda_{2}=\lambda_{3} \neq \lambda_{1}$, but there are still three linearly independent eigenvectors?

(iii) Suppose again that $\lambda_{2}=\lambda_{3} \neq \lambda_{1}$, but now there are only two linearly independent eigenvectors: $\mathbf{e}_{1}$ corresponding to $\lambda_{1}$ and $\mathbf{e}_{2}$ corresponding to $\lambda_{2}$. Suppose that a vector $\mathbf{v}$ satisfying the equation $\left(M-\lambda_{2} I\right) \mathbf{v}=\mathbf{e}_{2}$ exists, where $I$ denotes the identity matrix. Show that $\mathbf{v}$ is linearly independent of $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$, and hence or otherwise find the general solution of $(*)$.

Paper 2, Section II, B

commentSuppose that $u(x)$ satisfies the equation

$\frac{d^{2} u}{d x^{2}}-f(x) u=0$

where $f(x)$ is a given non-zero function. Show that under the change of coordinates $x=x(t)$,

$\frac{d^{2} u}{d t^{2}}-\frac{\ddot{x}}{\dot{x}} \frac{d u}{d t}-\dot{x}^{2} f(x) u=0$

where a dot denotes differentiation with respect to $t$. Furthermore, show that the function

$U(t)=\dot{x}^{-\frac{1}{2}} u(x)$

satisfies

$\frac{d^{2} U}{d t^{2}}-\left[\dot{x}^{2} f(x)+\dot{x}^{-\frac{1}{2}}\left(\frac{\ddot{x}}{\dot{x}} \frac{d}{d t}\left(\dot{x}^{\frac{1}{2}}\right)-\frac{d^{2}}{d t^{2}}\left(\dot{x}^{\frac{1}{2}}\right)\right)\right] U=0$

Choosing $\dot{x}=(f(x))^{-\frac{1}{2}}$, deduce that

$\frac{d^{2} U}{d t^{2}}-(1+F(t)) U=0$

for some appropriate function $F(t)$. Assuming that $F$ may be neglected, deduce that $u(x)$ can be approximated by

$u(x) \approx A(x)\left(c_{+} e^{G(x)}+c_{-} e^{-G(x)}\right),$

where $c_{+}, c_{-}$are constants and $A, G$ are functions that you should determine in terms of $f(x)$.

Paper 2, Section I, F

commentLet $A, B$ be events in the sample space $\Omega$ such that $0<P(A)<1$ and $0<P(B)<1$. The event $B$ is said to attract $A$ if the conditional probability $P(A \mid B)$ is greater than $P(A)$, otherwise it is said that $A$ repels $B$. Show that if $B$ attracts $A$, then $A$ attracts $B$. Does $B^{c}=\Omega \backslash B$ repel $A ?$

Paper 2, Section I, F

commentLet $U$ be a uniform random variable on $(0,1)$, and let $\lambda>0$.

(a) Find the distribution of the random variable $-(\log U) / \lambda$.

(b) Define a new random variable $X$ as follows: suppose a fair coin is tossed, and if it lands heads we set $X=U^{2}$ whereas if it lands tails we set $X=1-U^{2}$. Find the probability density function of $X$.

Paper 2, Section II, F

commentWhen coin $A$ is tossed it comes up heads with probability $\frac{1}{4}$, whereas coin $B$ comes up heads with probability $\frac{3}{4}$. Suppose one of these coins is randomly chosen and is tossed twice. If both tosses come up heads, what is the probability that coin $B$ was tossed? Justify your answer.

In each draw of a lottery, an integer is picked independently at random from the first $n$ integers $1,2, \ldots, n$, with replacement. What is the probability that in a sample of $r$ successive draws the numbers are drawn in a non-decreasing sequence? Justify your answer.

Paper 2, Section II, F

commentState and prove Markov's inequality and Chebyshev's inequality, and deduce the weak law of large numbers.

If $X$ is a random variable with mean zero and finite variance $\sigma^{2}$, prove that for any $a>0$,

$P(X \geqslant a) \leqslant \frac{\sigma^{2}}{\sigma^{2}+a^{2}}$

[Hint: Show first that $P(X \geqslant a) \leqslant P\left((X+b)^{2} \geqslant(a+b)^{2}\right)$ for every $b>0$.]

Paper 2, Section II, F

commentConsider the function

$\phi(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}, \quad x \in \mathbb{R}$

Show that $\phi$ defines a probability density function. If a random variable $X$ has probability density function $\phi$, find the moment generating function of $X$, and find all moments $E\left[X^{k}\right]$, $k \in \mathbb{N}$.

Now define

$r(x)=\frac{P(X>x)}{\phi(x)}$

Show that for every $x>0$,

$\frac{1}{x}-\frac{1}{x^{3}}<r(x)<\frac{1}{x}$

Paper 2, Section II, F

commentLionel and Cristiana have $a$ and $b$ million pounds, respectively, where $a, b \in \mathbb{N}$. They play a series of independent football games in each of which the winner receives one million pounds from the loser (a draw cannot occur). They stop when one player has lost his or her entire fortune. Lionel wins each game with probability $0<p<1$ and Cristiana wins with probability $q=1-p$, where $p \neq q$. Find the expected number of games before they stop playing.