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Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 7 a ii Solution by
Codex 0 2026-10-25
For an estimator of , its bias and mean squared error areWhen the second moment exists, expanding around gives the bias-variance decomposition .
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 7 a i Solution by
Codex 0 2026-10-25
For the exponential distribution with rate , the change of variables givesThe integrals are one and two by integration by parts. Hence .
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 6 b v Solution by
Codex 0 2026-10-25
By part iii, is an integer with probability zero. Thus almost surely, andThe pointwise inequalities implyPutting and rearranging yieldsThis extracts bounds for an infinite series from simple rounding inequalities.
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 6 b iv Solution by
Codex 0 2026-10-25
For a nonnegative integer-valued random variable , the tail-sum formula states . It follows by writing and using nonnegativity to interchange sum and expectation.
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 6 b iii Solution by
Codex 0 2026-10-25
For an integer , is the same event as . Integration for the Pareto distribution givesThe second identity subtracts two successive tail probabilities, equivalently integrating over . If includes zero, the extra case is and , since almost surely.
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 6 b ii Solution by
Codex 0 2026-10-25
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 6 b i Solution by
Codex 0 2026-10-25
Normalization of the probability density function requiresHence . The distribution is a Pareto distribution with minimum one and shape parameter .
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 6 a iii Solution by
Codex 0 2026-10-25
The integers form a countable set, and the events are disjoint. By countable additivity and part ii,Countability is essential: one cannot use the same sum argument over an arbitrary uncountable collection of zero-probability events.
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 6 a ii Solution by
Codex 0 2026-10-25
For a random variable with a probability density function, a singleton has zero probability:Equivalently, its cumulative distribution function is continuous, so it has no jump at .
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 6 a i Solution by
Codex 0 2026-10-25
The cumulative distribution function isThe total integral of the nonnegative probability density function is one.
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 5 b iii Solution by
Codex 0 2026-10-25
Put . ThenUniqueness of the probability generating function therefore givesThis is Bernoulli thinning of a geometric count, using the convention of counting failures before success.
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 5 b ii Solution by
Codex 0 2026-10-25
For an independent nonnegative integer-valued count and identically distributed independent summands, the random sum probability generating function is . Conditioning on proves this, since the conditional probability generating function is .
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 5 b i Solution by
Codex 0 2026-10-25
Take independent Bernoulli random variables of success probability , independently of . On the event , the random sum has the binomial distribution with parameters , including the empty sum zero at . It therefore has exactly the specified joint distribution with .
Thus one may realize on a suitable probability space. This is a distributional construction; it does not require a pre-existing to be independent of .
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 5 a iii Solution by
Codex 0 2026-10-25
Differentiate the probability generating function inside its radius of convergence:Since these are and , respectively,
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 5 a ii Solution by
Codex 0 2026-10-25
Writing , the probability generating function is a geometric series:The displayed restriction ensures absolute convergence.
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 5 a i Solution by
Codex 0 2026-10-25
With independent Bernoulli trials of success probability , counts the failures before the first success. There are failures followed by one success with probability , giving the stated geometric distribution on . It counts failures, rather than the total number of trials including the success.
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 4 b v Solution by
Codex 0 2026-10-25
The equilibrium of the first-order linear recurrence is . Subtracting it givesConsequentlyFor , interpret the formula at through the initial condition: , while for . For every , .
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 4 b iv Solution by
Codex 0 2026-10-25
Apply the law of total expectation to the partition of a sample space given by the possible values of . Since the conditional mean is affine,Here . This uses linearity of expectation and does not assert independence of successive urn counts.
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 4 b iii Solution by
Codex 0 2026-10-25
Past exam of the mathematics course of the University of Oxford 2023 Prelims Mathematics III 4 b ii Solution by
Codex 0 2026-10-25
Conditional on , urn 1 has red and blue balls, while urn 2 has red and blue balls. The independent selections therefore giveEvery other conditional probability is zero. The out-of-range terms at or already have zero probability. These are the transition probabilities of the Bernoulli-Laplace diffusion model.
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