• # Paper 2, Section I, A

Find the constant solutions (those with $u_{n+1}=u_{n}$ ) of the discrete equation

$u_{n+1}=\frac{1}{2} u_{n}\left(1+u_{n}\right)$

and determine their stability.

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• # Paper 2, Section I, A

Find two linearly independent solutions of

$y^{\prime \prime}+4 y^{\prime}+4 y=0 .$

Find the solution in $x \geqslant 0$ of

$y^{\prime \prime}+4 y^{\prime}+4 y=e^{-2 x},$

subject to $y=y^{\prime}=0$ at $x=0$.

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• # Paper 2, Section II, A

Consider the function

$V(x, y)=x^{4}-x^{2}+2 x y+y^{2}$

Find the critical (stationary) points of $V(x, y)$. Determine the type of each critical point. Sketch the contours of $V(x, y)=$ constant.

Now consider the coupled differential equations

$\frac{d x}{d t}=-\frac{\partial V}{\partial x}, \quad \frac{d y}{d t}=-\frac{\partial V}{\partial y}$

Show that $V(x(t), y(t))$ is a non-increasing function of $t$. If $x=1$ and $y=-\frac{1}{2}$ at $t=0$, where does the solution tend to as $t \rightarrow \infty$ ?

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• # Paper 2, Section II, A

Find the first three non-zero terms in the series solutions $y_{1}(x)$ and $y_{2}(x)$ for the differential equation

$x^{2} y^{\prime \prime}-2 x y^{\prime}+\left(2-x^{2}\right) y=0$

that satisfy

$\begin{array}{llc} y_{1}^{\prime}(0)=a & \text { and } & y_{1}^{\prime \prime}(0)=0 \\ y_{2}^{\prime}(0)=0 & \text { and } & y_{2}^{\prime \prime}(0)=2 b \end{array}$

Identify these solutions in closed form.

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• # Paper 2, Section II, A

Consider the second-order differential equation for $y(t)$ in $t \geqslant 0$

$\ddot{y}+2 k \dot{y}+\left(k^{2}+\omega^{2}\right) y=f(t) .$

(i) For $f(t)=0$, find the general solution $y_{1}(t)$ of $(*)$.

(ii) For $f(t)=\delta(t-a)$ with $a>0$, find the solution $y_{2}(t, a)$ of $(*)$ that satisfies $y=0$ and $\dot{y}=0$ at $t=0$.

(iii) For $f(t)=H(t-b)$ with $b>0$, find the solution $y_{3}(t, b)$ of $(*)$ that satisfies $y=0$ and $\dot{y}=0$ at $t=0 .$

(iv) Show that

$y_{2}(t, b)=-\frac{\partial y_{3}}{\partial b}$

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• # Paper 2, Section II, A

Find the solution to the system of equations

$\begin{gathered} \frac{d x}{d t}+\frac{-4 x+2 y}{t}=-9, \\ \frac{d y}{d t}+\frac{x-5 y}{t}=3 \end{gathered}$

in $t \geqslant 1$ subject to

$x=0 \quad \text { and } \quad y=0 \quad \text { at } \quad t=1$

[Hint: powers of t.]

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• # Paper 2, Section I, F

Define the probability generating function $G(s)$ of a random variable $X$ taking values in the non-negative integers.

A coin shows heads with probability $p \in(0,1)$ on each toss. Let $N$ be the number of tosses up to and including the first appearance of heads, and let $k \geqslant 1$. Find the probability generating function of $X=\min \{N, k\}$.

Show that $E(X)=p^{-1}\left(1-q^{k}\right)$ where $q=1-p$.

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• # Paper 2, Section I, F

Given two events $A$ and $B$ with $P(A)>0$ and $P(B)>0$, define the conditional probability $P(A \mid B)$.

Show that

$P(B \mid A)=P(A \mid B) \frac{P(B)}{P(A)}$

A random number $N$ of fair coins are tossed, and the total number of heads is denoted by $H$. If $P(N=n)=2^{-n}$ for $n=1,2, \ldots$, find $P(N=n \mid H=1)$.

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• # Paper 2, Section II, F

Let $X, Y$ be independent random variables with distribution functions $F_{X}, F_{Y}$. Show that $U=\min \{X, Y\}, V=\max \{X, Y\}$ have distribution functions

$F_{U}(u)=1-\left(1-F_{X}(u)\right)\left(1-F_{Y}(u)\right), \quad F_{V}(v)=F_{X}(v) F_{Y}(v)$

Now let $X, Y$ be independent random variables, each having the exponential distribution with parameter 1. Show that $U$ has the exponential distribution with parameter 2 , and that $V-U$ is independent of $U$.

Hence or otherwise show that $V$ has the same distribution as $X+\frac{1}{2} Y$, and deduce the mean and variance of $V$.

[You may use without proof that $X$ has mean 1 and variance 1.]

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• # Paper 2, Section II, F

(i) Let $X_{n}$ be the size of the $n^{\text {th }}$generation of a branching process with familysize probability generating function $G(s)$, and let $X_{0}=1$. Show that the probability generating function $G_{n}(s)$ of $X_{n}$ satisfies $G_{n+1}(s)=G\left(G_{n}(s)\right)$ for $n \geqslant 0$.

(ii) Suppose the family-size mass function is $P\left(X_{1}=k\right)=2^{-k-1}, k=0,1,2, \ldots$ Find $G(s)$, and show that

$G_{n}(s)=\frac{n-(n-1) s}{n+1-n s} \quad \text { for }|s|<1+\frac{1}{n} .$

Deduce the value of $P\left(X_{n}=0\right)$.

(iii) Write down the moment generating function of $X_{n} / n$. Hence or otherwise show that, for $x \geqslant 0$,

$P\left(X_{n} / n>x \mid X_{n}>0\right) \rightarrow e^{-x} \quad \text { as } n \rightarrow \infty$

[You may use the continuity theorem but, if so, should give a clear statement of it.]

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• # Paper 2, Section II, F

(i) Define the distribution function $F$ of a random variable $X$, and also its density function $f$ assuming $F$ is differentiable. Show that

$f(x)=-\frac{d}{d x} P(X>x)$

(ii) Let $U, V$ be independent random variables each with the uniform distribution on $[0,1]$. Show that

$P\left(V^{2}>U>x\right)=\frac{1}{3}-x+\frac{2}{3} x^{3 / 2}, \quad x \in(0,1)$

What is the probability that the random quadratic equation $x^{2}+2 V x+U=0$ has real roots?

Given that the two roots $R_{1}, R_{2}$ of the above quadratic are real, what is the probability that both $\left|R_{1}\right| \leqslant 1$ and $\left|R_{2}\right| \leqslant 1 ?$

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• # Paper 2, Section II, F

(i) Define the moment generating function $M_{X}(t)$ of a random variable $X$. If $X, Y$ are independent and $a, b \in \mathbb{R}$, show that the moment generating function of $Z=a X+b Y$ is $M_{X}(a t) M_{Y}(b t)$.

(ii) Assume $T>0$, and $M_{X}(t)<\infty$ for $|t|. Explain the expansion

$M_{X}(t)=1+\mu t+\frac{1}{2} s^{2} t^{2}+\mathrm{o}\left(t^{2}\right)$

where $\mu=E(X)$ and $s^{2}=E\left(X^{2}\right) . \quad$ [You may assume the validity of interchanging expectation and differentiation.]

(iii) Let $X, Y$ be independent, identically distributed random variables with mean 0 and variance 1 , and assume their moment generating function $M$ satisfies the condition of part (ii) with $T=\infty$.

Suppose that $X+Y$ and $X-Y$ are independent. Show that $M(2 t)=M(t)^{3} M(-t)$, and deduce that $\psi(t)=M(t) / M(-t)$ satisfies $\psi(t)=\psi(t / 2)^{2}$.

Show that $\psi(h)=1+\mathrm{o}\left(h^{2}\right)$ as $h \rightarrow 0$, and deduce that $\psi(t)=1$ for all $t$.

Show that $X$ and $Y$ are normally distributed.

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