• # 2.I.1B

Solve the equation

$\frac{d y}{d x}+3 x^{2} y=x^{2}$

with $y(0)=a$, by use of an integrating factor or otherwise. Find $\lim _{x \rightarrow+\infty} y(x)$.

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• # 2.I.2B

Obtain the general solution of

$x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=0$

by using the indicial equation.

Introduce $z=\log x$ as a new independent variable and find an equivalent second order differential equation with constant coefficients. Determine the general solution of this new equation, and show that it is equivalent to the general solution of $(*)$ found previously.

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• # 2.II.5B

Find two linearly independent solutions of the difference equation

$X_{n+2}-2 \cos \theta X_{n+1}+X_{n}=0$

for all values of $\theta \in(0, \pi)$. What happens when $\theta=0$ ? Find two linearly independent solutions in this case.

Find $X_{n}(\theta)$ which satisfy the initial conditions

$X_{1}=1, \quad X_{2}=2,$

for $\theta=0$ and for $\theta \in(0, \pi)$. For every $n$, show that $X_{n}(\theta) \rightarrow X_{n}(0)$ as $\theta \rightarrow 0$.

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• # 2.II.7B

The Cartesian coordinates $(x, y)$ of a point moving in $\mathbb{R}^{2}$ are governed by the system

\begin{aligned} &\frac{d x}{d t}=-y+x\left(1-x^{2}-y^{2}\right), \\ &\frac{d y}{d t}=x+y\left(1-x^{2}-y^{2}\right) . \end{aligned}

Transform this system of equations to polar coordinates $(r, \theta)$ and hence find all periodic solutions (i.e., closed trajectories) which satisfy $r=$ constant.

Discuss the large time behaviour of an arbitrary solution starting at initial point $\left(x_{0}, y_{0}\right)=\left(r_{0} \cos \theta_{0}, r_{0} \sin \theta_{0}\right)$. Summarize the motion using a phase plane diagram, and comment on the nature of any critical points.

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• # 2.II.8B

Define the Wronskian $W\left[u_{1}, u_{2}\right]$ for two solutions $u_{1}, u_{2}$ of the equation

$\frac{d^{2} u}{d x^{2}}+p(x) \frac{d u}{d x}+q(x) u=0$

and obtain a differential equation which exhibits its dependence on $x$. Explain the relevance of the Wronskian to the linear independence of $u_{1}$ and $u_{2}$.

Consider the equation

$x^{2} \frac{d^{2} y}{d x^{2}}-2 y=0$

and determine the dependence on $x$ of the Wronskian $W\left[y_{1}, y_{2}\right]$ of two solutions $y_{1}$ and $y_{2}$. Verify that $y_{1}(x)=x^{2}$ is a solution of $(*)$ and use the Wronskian to obtain a second linearly independent solution.

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• # 2.I.3F

Suppose $c \geqslant 1$ and $X_{c}$ is a positive real-valued random variable with probability density

$f_{c}(t)=A_{c} t^{c-1} e^{-t^{c}},$

for $t>0$, where $A_{c}$ is a constant.

Find the constant $A_{c}$ and show that, if $c>1$ and $s, t>0$,

$\mathbb{P}\left[X_{c} \geqslant s+t \mid X_{c} \geqslant t\right]<\mathbb{P}\left[X_{c} \geqslant s\right]$

[You may assume the inequality $(1+x)^{c}>1+x^{c}$ for all $x>0, c>1$.]

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• # 2.I.4F

Describe the Poisson distribution characterised by parameter $\lambda>0$. Calculate the mean and variance of this distribution in terms of $\lambda$.

Show that the sum of $n$ independent random variables, each having the Poisson distribution with $\lambda=1$, has a Poisson distribution with $\lambda=n$.

Use the central limit theorem to prove that

$e^{-n}\left(1+\frac{n}{1 !}+\frac{n^{2}}{2 !}+\ldots+\frac{n^{n}}{n !}\right) \rightarrow 1 / 2 \quad \text { as } \quad n \rightarrow \infty$

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• # 2.II $. 9 \mathrm{~F} \quad$

Given a real-valued random variable $X$, we define $\mathbb{E}\left[e^{i X}\right]$ by

$\mathbb{E}\left[e^{i X}\right] \equiv \mathbb{E}[\cos X]+i \mathbb{E}[\sin X]$

Consider a second real-valued random variable $Y$, independent of $X$. Show that

$\mathbb{E}\left[e^{i(X+Y)}\right]=\mathbb{E}\left[e^{i X}\right] \mathbb{E}\left[e^{i Y}\right]$

You gamble in a fair casino that offers you unlimited credit despite your initial wealth of 0 . At every game your wealth increases or decreases by $£ 1$ with equal probability $1 / 2$. Let $W_{n}$ denote your wealth after the $n^{t h}$ game. For a fixed real number $u$, compute $\phi(u)$ defined by

$\phi(u)=\mathbb{E}\left[e^{i u W_{n}}\right]$

Verify that the result is real-valued.

Show that for $n$ even,

$\mathbb{P}\left[W_{n}=0\right]=\gamma \int_{0}^{\pi / 2}[\cos u]^{n} d u$

for some constant $\gamma$, which you should determine. What is $\mathbb{P}\left[W_{n}=0\right]$ for $n$ odd?

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• # 2.II.10F

Alice and Bill fight a paint-ball duel. Nobody has been hit so far and they are both left with one shot. Being exhausted, they need to take a breath before firing their last shot. This takes $A$ seconds for Alice and $B$ seconds for Bill. Assume these times are exponential random variables with means $1 / \alpha$ and $1 / \beta$, respectively.

Find the distribution of the (random) time that passes by before the next shot is fired. What is its standard deviation? What is the probability that Alice fires the next shot?

Assume Alice has probability $1 / 2$ of hitting whenever she fires whereas Bill never misses his target. If the next shot is a hit, what is the probability that it was fired by Alice?

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• # 2.II.11F

Let $(S, T)$ be uniformly distributed on $[-1,1]^{2}$ and define $R=\sqrt{S^{2}+T^{2}}$. Show that, conditionally on

$R \leqslant 1,$

the vector $(S, T)$ is uniformly distributed on the unit disc. Let $(R, \Theta)$ denote the point $(S, T)$ in polar coordinates and find its probability density function $f(r, \theta)$ for $r \in[0,1], \theta \in[0,2 \pi)$. Deduce that $R$ and $\Theta$ are independent.

Introduce the new random variables

$X=\frac{S}{R} \sqrt{-2 \log \left(R^{2}\right)}, \quad Y=\frac{T}{R} \sqrt{-2 \log \left(R^{2}\right)}$

noting that under the above conditioning, $(S, T)$ are uniformly distributed on the unit disc. The pair $(X, Y)$ may be viewed as a (random) point in $\mathbb{R}^{2}$ with polar coordinates $(Q, \Psi)$. Express $Q$ as a function of $R$ and deduce its density. Find the joint density of $(Q, \Psi)$. Hence deduce that $X$ and $Y$ are independent normal random variables with zero mean and unit variance.

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• # 2.II.12F

Let $a_{1}, a_{2}, \ldots, a_{n}$ be a ranking of the yearly rainfalls in Cambridge over the next $n$ years: assume $a_{1}, a_{2}, \ldots, a_{n}$ is a random permutation of $1,2, \ldots, n$. Year $k$ is called a record year if $a_{i}>a_{k}$ for all $i (thus the first year is always a record year). Let $Y_{i}=1$ if year $i$ is a record year and 0 otherwise.

Find the distribution of $Y_{i}$ and show that $Y_{1}, Y_{2}, \ldots, Y_{n}$ are independent and calculate the mean and variance of the number of record years in the next $n$ years.

Find the probability that the second record year occurs at year $i$. What is the expected number of years until the second record year occurs?

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