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MATHEMATICAL TRIPOS Part IA
Friday 31 May 2002 1.30 to 4.
PAPER 2
Before you begin read these instructions carefully.
Each question in Section II carries twice the credit of each question in Section I. You may attempt all four questions in Section I. In Section II at most five answers will be taken into account and no more than three answers on each course will be taken into account.
Complete answers are preferred to fragments.
Write on one side of the paper only and begin each answer on a separate sheet.
Write legibly; otherwise you place yourself at a grave disadvantage.
At the end of the examination:
Tie up your answers in two bundles, marked D and F according to the code letter affixed to each question. Attach a blue cover sheet to each bundle; write the code in the box marked ‘SECTION’ on the cover sheet. Do not tie up questions from Section I and Section II in separate bundles.
You must also complete a green master cover sheet listing all the questions attempted by you.
Every cover sheet must bear your examination number and desk number.
SECTION I
1D Differential Equations
Solve the equation y¨ + ˙y − 2 y = e−t
subject to the conditions y(t) = ˙y(t) = 0 at t = 0. Solve the equation
y¨ + ˙y − 2 y = et
subject to the same conditions y(t) = ˙y(t) = 0 at t = 0.
2D Differential Equations Consider the equation
dy dx
= x
1 − y^2 1 − x^2
where the positive square root is taken, within the square S : 0 6 x < 1 , 0 6 y 6 1. Find the solution that begins at x = y = 0. Sketch the corresponding solution curve, commenting on how its tangent behaves near each extremity. By inspection of the right- hand side of (∗), or otherwise, roughly sketch, using small line segments, the directions of flow throughout the square S.
3F Probability Define the indicator function IA of an event A.
Let Ii be the indicator function of the event Ai, 1 ≤ i ≤ n, and let N =
∑n 1 Ii^ be the number of values of i such that Ai occurs. Show that E(N ) =
i pi^ where^ pi^ =^ P^ (Ai), and find var(N ) in terms of the quantities pij = P (Ai ∩ Aj ).
Using Chebyshev’s inequality or otherwise, show that
P (N = 0) ≤
var(N ) {E(N )}^2
4F Probability
A coin shows heads with probability p on each toss. Let πn be the probability that the number of heads after n tosses is even. Show carefully that πn+1 = (1−p)πn +p(1−πn), n ≥ 1, and hence find πn. [The number 0 is even.]
Paper 2
6D Differential Equations
Solve the differential equation
dy dt
= ry (1 − ay)
for the general initial condition y = y 0 at t = 0, where r, a, and y 0 are positive constants. Deduce that the equilibria at y = a−^1 and y = 0 are stable and unstable, respectively.
By using the approximate finite-difference formula
dy dt
yn+1 − yn δt
for the derivative of y at t = nδt, where δt is a positive constant and yn = y(n δt), show that the differential equation when thus approximated becomes the difference equation
un+1 = λ (1 − un) un ,
where λ = 1 + r δt > 1 and where un = λ−^1 a(λ − 1) yn. Find the two equilibria and, by linearizing the equation about them or otherwise, show that one is always unstable (given that λ > 1) and that the other is stable or unstable according as λ < 3 or λ > 3. Show that this last instability is oscillatory with period 2 δt. Why does this last instability have no counterpart for the differential equation? Show graphically how this instability can equilibrate to a periodic, finite-amplitude oscillation when λ = 3.2.
Paper 2
7D Differential Equations
The homogeneous equation
y¨ + p(t) ˙y + q(t)y = 0
has non-constant, non-singular coefficients p(t) and q(t). Two solutions of the equation, y(t) = y 1 (t) and y(t) = y 2 (t), are given. The solutions are known to be such that the determinant
W (t) =
∣∣ y^1 y^2 y ˙ 1 y˙ 2
is non-zero for all t. Define what is meant by linear dependence, and show that the two given solutions are linearly independent. Show also that
W (t) ∝ exp
∫ (^) t p(s) ds
In the corresponding inhomogeneous equation
y¨ + p(t) ˙y + q(t)y = f (t)
the right-hand side f (t) is a prescribed forcing function. Construct a particular integral of this inhomogeneous equation in the form
y(t) = a 1 (t) y 1 (t) + a 2 (t) y 2 (t) ,
where the two functions ai(t) are to be determined such that
y 1 (t) ˙a 1 (t) + y 2 (t) ˙a 2 (t) = 0
for all t. Express your result for the functions ai(t) in terms of integrals of the functions f (t) y 1 (t)/W (t) and f (t) y 2 (t)/W (t).
Consider the case in which p(t) = 0 for all t and q(t) is a positive constant, q = ω^2 say, and in which the forcing f (t) = sin(ωt). Show that in this case y 1 (t) and y 2 (t) can be taken as cos(ωt) and sin(ωt) respectively. Evaluate f (t) y 1 (t)/W (t) and f (t) y 2 (t)/W (t) and show that, as t → ∞, one of the ai(t) increases in magnitude like a power of t to be determined.
Paper 2 [TURN OVER
10F Probability
There is a random number N of foreign objects in my soup, with mean μ and finite variance. Each object is a fly with probability p, and otherwise is a spider; different objects have independent types. Let F be the number of flies and S the number of spiders.
(a) Show that GF (s) = GN (ps + 1 − p). [GX denotes the probability generating function of a random variable X. You should present a clear statement of any general result used.]
(b) Suppose N has the Poisson distribution with parameter μ. Show that F has the Poisson distribution with parameter μp, and that F and S are independent.
(c) Let p = 12 and suppose that F and S are independent. [You are given nothing about the distribution of N .] Show that GN (s) = GN ( 12 (1 + s))^2. By working with the function H(s) = GN (1 − s) or otherwise, deduce that N has the Poisson distribution. [You may assume that
1 + xn + o(n−^1 )
)n → ex^ as n → ∞.]
11F Probability
Let X, Y , Z be independent random variables each with the uniform distribution on the interval [0, 1].
(a) Show that X + Y has density function
fX+Y (u) =
{ (^) u if 0 ≤ u ≤ 1, 2 − u if 1 ≤ u ≤ 2, 0 otherwise.
(b) Show that P (Z > X + Y ) = 16.
(c) You are provided with three rods of respective lengths X, Y , Z. Show that the probability that these rods may be used to form the sides of a triangle is 12.
(d) Find the density function fX+Y +Z (s) of X + Y + Z for 0 6 s 6 1. Let W be uniformly distributed on [0, 1], and independent of X, Y , Z. Show that the probability that rods of lengths W , X, Y , Z may be used to form the sides of a quadrilateral is 56.
Paper 2 [TURN OVER
12F Probability
(a) Explain what is meant by the term ‘branching process’.
(b) Let Xn be the size of the nth generation of a branching process in which each family size has probability generating function G, and assume that X 0 = 1. Show that the probability generating function Gn of Xn satisfies Gn+1(s) = Gn(G(s)) for n ≥ 1.
(c) Show that G(s) = 1 − α(1 − s)β^ is the probability generating function of a non-negative integer-valued random variable when α, β ∈ (0, 1), and find Gn explicitly when G is thus given.
(d) Find the probability that Xn = 0, and show that it converges as n → ∞ to 1−α^1 /(1−β). Explain carefully why this implies that the probability of ultimate extinction equals 1 − α^1 /(1−β).
END OF PAPER
Paper 2
