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MATHEMATICAL TRIPOS Part IA
Friday 1st June 2007 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 B 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 gold cover sheet to each bundle; write the code letter in the box marked ‘EXAMINER LETTER’ on the cover sheet.
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 sheet 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
1B Differential Equations
Find the solution y(x) of the equation
y′′^ − 6 y′^ + 9y = cos(2x) e^3 x
that satisfies y(0) = 0 and y′(0) = 1.
2B Differential Equations
Investigate the stability of: (i) the equilibrium points of the equation
dy dt
= (y^2 − 4) tan−^1 (y) ;
(ii) the constant solutions (un+1 = un) of the discrete equation
un+1 =
u^2 n(1 + un).
3F Probability
Let X and Y be independent random variables, each uniformly distributed on [0, 1]. Let U = min(X, Y ) and V = max(X, Y ). Show that EU = 13 , and hence find the covariance of U and V.
4F Probability
Let X be a normally distributed random variable with mean 0 and variance 1. Define, and determine, the moment generating function of X. Compute EXr^ for r = 0, 1 , 2 , 3 , 4.
Let Y be a normally distributed random variable with mean μ and variance σ^2. Determine the moment generating function of Y.
Paper 2
7B Differential Equations
(i) Find, in the form of an integral, the solution of the equation
α
dy dt
that satisfies y → 0 as t → −∞. Here f (t) is a general function and α is a positive constant.
Hence find the solution in each of the cases: (a) f (t) = δ(t) ;
(b) f (t) = H(t), where H(t) is the Heaviside step function. (ii) Find and sketch the solution of the equation
dy dt
given that y(0) = 0 and y(t) is continuous.
8B Differential Equations
(i) Find the general solution of the difference equation
uk+1 + 5uk + 6uk− 1 = 12k + 1.
(ii) Find the solution of the equation
yk+1 + 5yk + 6yk− 1 = 2k
that satisfies y 0 = y 1 = 1. Hence show that, for any positive integer n, the quantity 2 n^ − 26(−3)n^ is divisible by 10.
Paper 2
9F Probability
Let N be a non-negative integer-valued random variable with
P {N = r} = pr , r = 0, 1 , 2 ,...
Define EN , and show that
EN =
∑^ ∞
n=
P {N > n}.
Let X 1 , X 2 ,... be a sequence of independent and identically distributed continuous random variables. Let the random variable N mark the point at which the sequence stops decreasing: that is, N > 2 is such that
X 1 > X 2 >... > XN − 1 < XN ,
where, if there is no such finite value of N , we set N = ∞. Compute P {N = r}, and show that P {N = ∞} = 0. Determine EN.
10F Probability
Let X and Y be independent non-negative random variables, with densities f and g respectively. Find the joint density of U = X and V = X + aY , where a is a positive constant.
Let X and Y be independent and exponentially distributed random variables, each with density f (x) = λe−λx, x > 0.
Find the density of X + 12 Y. Is it the same as the density of the random variable max(X, Y )?
Paper 2 [TURN OVER
