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This is the Exam of Statistical Science which includes Nineteenth Century, Data, First Few Lines, Subset, Subregions, West and Central, Department Poplulation, Number, Crimes Against Property etc. Key important points are:Mean Square, Random Variables, Distribution, Almost Surely, Distribution, Independent, Increasing Sequence, Borel Cantelli Lemma, Natural Numbers, Independent Exponential Random Variables
Typology: Exams
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Thursday 1 June 2006 9-
Attempt THREE questions. There are FIVE questions in total. Marks for each question are indicated on the paper in square brackets. Each question is worth a total of 20 marks.
Cover sheet None Treasury Tag Script paper
1 Let X 1 , X 2 ,... and X be random variables.
(a) What does it mean to say
(i) Xn → X in distribution? (ii) Xn → X almost surely?
(iii) Xn → X in L 2 (or mean-square)? (iv) Xn → X in probability? [6]
(b) Prove that Xn → 0 in distribution if and only if Xn → 0 in probability. [4]
(c) Let Y 1 , Y 2 ,... be independent with E(Yi) = μ and Var(Yi) = σ^2 for all
i = 1, 2 ,.. .. Prove that
Y 1 +... + Yn n
→ μ in probability. [4]
(d) Let Xn → X in probability. Use the first Borel-Cantelli lemma to prove that there exists an increasing sequence k 1 , k 2 ,... of natural numbers such that Xkn → X almost surely. [6]
2 Let X 1 , X 2 ,... be independent exponential random variables with rate parameter λ (so that E(Xi) = 1/λ for all i = 1, 2 ,.. .)
Let N be a Poisson random variable with intensity parameter μ and let U be a random variable uniformly distributed on [0, 1]. Suppose that N, U, X 1 , X 2 ,... are independent. Let Y = X 1 +... + XN and Z = (X 1 + X 2 )U.
(a) Prove that the moment generating function of Y is MY (t) = exp
μt λ−t
for
t < λ. [6]
(b) Prove that Z is an exponential random variable with rate λ. [6] (c) Find the conditional density of X 1 given that X 1 + X 2 = 1. [8]
Introduction to Probability
5 Let G = (gij )i,j> 0 be a matrix with entries
gij =
λ if j = i + 1 −λ if j = i = 0 −(μ + λ) if j = i > 1 μ if j = i − 1
for i, j = 0, 1 , 2 ,... and constants λ, μ > 0.
Let (Xt)t> 0 be a continuous time Markov chain with generator G.
(a) Prove that if λ < μ there exists a unique invariant measure for the chain. Find it. [8]
(b) What are the forward Kolmogorov equations for the transition probabilities pij (t) = P(Xt = j|X 0 = i)? [6]
(c) Prove that if μ = 0 and X 0 = 0 then the random variable Xt has the Poisson distribution with parameter λt. [6]
Introduction to Probability
