M. PHIL. IN STATISTICAL SCIENCE
Friday 1 June 2001 9 to 11
PROBABILITY
Attempt three of the following five questions.
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
Prepare for your exams
Study with the several resources on Docsity
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Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Probability - Statistical Science - Exam, Exams of Statistics
This is the Exam of Statistical Science which includes Recursive Method, Time Series, Observations, Stationary Autoregressive Process, Obtaining Forecasts, Noise Process, Weakly Stationary, Considering, Recursive Forecasts etc. Key important points are: Probability, Mean, Transition Matrix, Markov Property, Infinitely, Non Zero Transition Probabilities, Sequence of Independent, Identically Distributed Random, True, Reference to Standard Theorems
Typology: Exams
2012/2013
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M. PHIL. IN STATISTICAL SCIENCE
Friday 1 June 2001 9 to 11
PROBABILITY
Attempt three of the following five questions.
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
(^1) (a) What does it mean to say that (Xn)n> 0 is a discrete-time Markov chain with state space S and transition matrix P?
(b) State the strong Markov property of such a process (Xn)n> 0. (c) Fix a state i and let fi denote the return probability for i, that is, the probability when X 0 = i, that Xn = i for some n > 1. Show that, if fi = 1, then Xn visits i infinitely often, or not at all.
Show also that, if fi < 1, then Xn visits i only finitely often.
(d) Consider the Markov chain (Xn)n> 0 in Z whose non-zero transition probabilities are given by P 0 ,− 1 = 1, Pi,i− 1 =
, Pi,i+1 =
, i 6 = 0.
If X 0 = 1, what is the probability that Xn visits 0 infinitely often.
2 Let X, X 1 , X 2 ,... be a sequence of independent, identically distributed random variables. Suppose that E(X) = μ, E(X^4 ) < ∞.
(a) Show that X 1 +... + Xn n
→ μ a.s. as n → ∞
(b) Suppose that μ = 0. Is it true that (X 1 +... + Xn)/
n converges in distribution as n → ∞? Is it true that (X 1 +... + Xn)/
n converges in L 2 (P)? Justify your answers by reference to standard theorems.
PROBABILITY
4 Let X, X 1 , X 2 ,... be independent, identically distributed random variables.
Set MX (t) = E(etX^ ), cX (t) = log MX (t).
(a) Show, for Sn = X 1 +... + Xn, for t > 0 and x ∈ R, that
P(Sn > nx) 6 e−n{xt−cX^ (t)}.
Show also that this inequality remains valid for t < 0, provided x > E(X).
Deduce that, for x > E(X),
lim sup n→∞
n
log P(
Sn n
x) 6 −IX (x),
where IX (x) = sup t∈R
{xt − cX (t)}.
(b) Write Iλ for IX when X ∼ E(λ) and write Ip for Ix when X ∼ B(1, p). Compute Iλ and Ip. Here, E(λ) denotes the exponential distribution of rate λ and B(n, p) denotes the binomial distribution with parameters n and p.
(c) When fitting a light bulb, there is a chance of 1 − p that it fails instantaneously. Given that it does not, its lifetime has E(λ) distribution. Let pn(x) denote the probability that the average lifetime of n bulbs exceeds x, where x > 1 /λ. Show that, for all n, pn(x) 6 E(e−nFx(N/n)) where N has B(n, p) distribution and
Fx(y) = yIλ(x/y).
Deduce that
lim sup n→∞
n
log pn(x) 6 − inf y∈[0,1]
{Ip(y) + yIλ(x/y)}.
5 State Doob’s L 1 and L 2 martingale convergence theorems.
Let f : [0, 1] → R and assume that for some K > 0, we have |f (x)−f (y)| 6 K|x−y| for all x and y. For every n, let ti,n = i/ 2 n. Define Mn : [0, 1) → R by
Mn(x) = 2n(f (ti+1,n) − f (ti,n)), ti,n 6 x < ti+1,n, i = 0, 1 ,... , 2 n^ − 1.
Show that, for almost all x, the limit M (x) = limn→∞ Mn(x) exists and satisfies ∫ (^1)
0
M (x)dx = f (1) − f (0).
What is M better known as?