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MATHEMATICAL TRIPOS Part IA
Friday, 29 May, 2009 1:30 pm to 4:30 pm
PAPER 2
Before you begin read these instructions carefully.
The examination paper is divided into two sections. Each question in Section II carries twice the number of marks of each question in Section I. Candidates may attempt all four questions from Section I and at most five questions from Section II. In Section II, no more than three questions on each course may be attempted.
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 separate bundles, marked C and F according to the code letter affixed to each question. Include in the same bundle all questions from Section I and II with the same code letter.
Attach a completed gold cover sheet to each bundle.
You must also complete a green master cover sheet listing all the questions you have attempted.
Every cover sheet must bear your examination number and desk number.
STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS
Gold cover sheets None Green master cover sheet
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
SECTION I
1C Differential Equations The size of the population of ducks living on the pond of a certain Cambridge college is governed by the equation dN dt = αN − N 2 ,
where N = N (t) is the number of ducks at time t and α is a positive constant. Given that N (0) = 2α, find N (t). What happens as t → ∞?
2C Differential Equations Solve the differential equation
d^2 y dx^2
dy dx
subject to the conditions y = dy/dx = 0 when x = 0.
3F Probability Consider a pair of jointly normal random variables X 1 , X 2 , with mean values μ 1 , μ 2 , variances σ 12 , σ 22 and correlation coefficient ρ with |ρ| < 1.
(a) Write down the joint probability density function for (X 1 , X 2 ).
(b) Prove that X 1 , X 2 are independent if and only if ρ = 0.
4F Probability Prove the law of total probability: if A 1 ,.. ., An are pairwise disjoint events with
P(Ai) > 0, and B ⊆ A 1 ∪... ∪ An then P(B) =
∑n i=
P(Ai)P(B|Ai).
There are n people in a lecture room. Their birthdays are independent random variables, and each person’s birthday is equally likely to be any of the 365 days of the year. By using the bound 1 − x 6 e−x^ for 0 6 x 6 1, prove that if n > 29 then the probability that at least two people have the same birthday is at least 2/3.
[In calculations, you may take
1 + 8 × 365 ln 3 = 56. 6 .]
Part IA, Paper 2
7C Differential Equations Consider the differential equation
x d^2 y dx^2
where c is a constant with 0 < c < 1. Determine two linearly independent series solutions about x = 0, giving an explicit expression for the coefficient of the general term in each series.
Determine the solution of
x d^2 y dx^2
for which y(0) = 0 and dy/dx is finite at x = 0.
8C Differential Equations (a) The function y(x, t) satisfies the forced wave equation
∂^2 y ∂x^2
∂^2 y ∂t^2
with initial conditions y(x, 0) = sin x and ∂y/∂t(x, 0) = 0. By making the change of variables u = x + t and v = x − t, show that
∂^2 y ∂u∂v
Hence, find y(x, t).
(b) The thickness of an axisymmetric drop of liquid spreading on a flat surface satisfies
∂h ∂t
r
∂r
rh^3 ∂h ∂r
where h = h(r, t) is the thickness of the drop, r is the radial coordinate on the surface and t is time. The drop has radius R(t). The boundary conditions are that ∂h/∂r = 0 at r = 0 and h(r, t) ∝ (R(t) − r)^1 /^3 as r → R(t).
Show that M =
∫ (^) R(t)
0
rh dr
is independent of time. Given that h(r, t) = f (r/tα)t−^1 /^4 for some function f (which need not be determined) and that R(t) is proportional to tα, find α.
Part IA, Paper 2
9F Probability I throw two dice and record the scores S 1 and S 2. Let X be the sum S 1 + S 2 and Y the difference S 1 − S 2.
(a) Suppose that the dice are fair, so the values 1,... , 6 are equally likely. Calculate the mean and variance of both X and Y. Find all the values of x and y at which the probabilities P(X = x), P(Y = y) are each either greatest or least. Determine whether the random variables X and Y are independent.
(b) Now suppose that the dice are unfair, and that they give the values 1,... , 6 with probabilities p 1 ,... , p 6 and q 1 ,... , q 6 , respectively. Write down the values of P(X = 2), P(X = 7) and P(X = 12). By comparing P(X = 7) with
P(X = 2) P(X = 12)
and applying the arithmetic-mean–geometric-mean inequality, or otherwise, show that the probabilities P(X = 2), P(X = 3),.. ., P(X = 12) cannot all be equal.
Part IA, Paper 2 [TURN OVER
11F Probability Let X and Y be two independent uniformly distributed random variables on [0, 1]. Prove that EXk^ =
k + 1
and E(XY )k^ =
(k + 1)^2
, and find E(1 − XY )k, where k is a non-negative integer. Let (X 1 , Y 1 ),... , (Xn, Yn) be n independent random points of the unit square S = {(x, y) : 0 6 x, y 6 1 }. We say that (Xi, Yi) is a maximal external point if, for each j = 1,... , n, either Xj 6 Xi or Yj 6 Yi. (For example, in the figure below there are three maximal external points.) Determine the expected number of maximal external points.
..^.
Part IA, Paper 2 [TURN OVER
12F Probability Let A 1 , A 2 and A 3 be three pairwise disjoint events such that the union A 1 ∪A 2 ∪A 3 is the full event and P(A 1 ), P(A 2 ), P(A 3 ) > 0. Let E be any event with P(E) > 0. Prove the formula
P(Ai|E) = ∑P(Ai)P(E|Ai) j=1, 2 , 3
P(Aj )P(E|Aj )
A Royal Navy speedboat has intercepted an abandoned cargo of packets of the deadly narcotic spitamin. This sophisticated chemical can be manufactured in only three places in the world: a plant in Authoristan (A), a factory in Bolimbia (B) and the ultramodern laboratory on board of a pirate submarine Crash (C) cruising ocean waters. The investigators wish to determine where this particular cargo comes from, but in the absence of prior knowledge they have to assume that each of the possibilities A, B and C is equally likely.
It is known that a packet from A contains pure spitamin in 95% of cases and is contaminated in 5% of cases. For B the corresponding figures are 97% and 3%, and for C they are 99% and 1%.
Analysis of the captured cargo showed that out of 10000 packets checked, 9800 contained the pure drug and the remaining 200 were contaminated. On the basis of this analysis, the Royal Navy captain estimated that 98% of the packets contain pure spitamin and reported his opinion that with probability roughly 0.5 the cargo was produced in B and with probability roughly 0.5 it was produced in C.
Assume that the number of contaminated packets follows the binomial distribution Bin(10000, δ/100) where δ equals 5 for A, 3 for B and 1 for C. Prove that the captain’s opinion is wrong: there is an overwhelming chance that the cargo comes from B.
[Hint: Let E be the event that 200 out of 10000 packets are contaminated. Compare the ratios of the conditional probabilities P(E|A), P(E|B) and P(E|C). You may find it helpful that ln 3 ≈ 1. 09861 and ln 5 ≈ 1. 60944. You may also take ln(1−δ/100) ≈ −δ/ 100 .]
END OF PAPER
Part IA, Paper 2
