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Past exam of the mathematics course of the University of Cambridge 2020 ii Paper 4 31A iii by
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Past exam of the mathematics course of the University of Cambridge 2020 ii Paper 4 31A ii by
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Past exam of the mathematics course of the University of Cambridge 2020 ii Paper 4 31A i by
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Past exam of the mathematics course of the University of Cambridge 2020 ii Paper 4 30J iv by
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Past exam of the mathematics course of the University of Cambridge 2020 ii Paper 4 30J iii by
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Past exam of the mathematics course of the University of Cambridge 2020 ii Paper 4 30J ii by
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Past exam of the mathematics course of the University of Cambridge 2020 ii Paper 4 2H Solution by
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A subset of a metric space is nowhere dense when the interior of its closure is empty. One form of the Baire category theorem says that a nonempty complete metric space cannot be a countable union of nowhere dense closed sets; equivalently, a countable intersection of dense open sets is dense.
Fix and work in the complete interval . For define the closed setThe sets are closed by the continuity of , and the assumed pointwise convergence gives . The Baire category theorem therefore gives an and a nonempty open interval contained in .
The intervals overlap once , which holds for every sufficiently large natural number . Their union consequently contains a half-line. Every sufficiently large can therefore be written as with and , and then . Since was arbitrary, as .
Past exam of the mathematics course of the University of Cambridge 2020 ii Paper 4 29K v by
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Past exam of the mathematics course of the University of Cambridge 2020 ii Paper 4 29K iii by
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Past exam of the mathematics course of the University of Cambridge 2020 ii Paper 4 28J Solution by
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Write and let be its score function. A second-order Taylor expansion around gives, for fixed ,where every lies between and . The mean-zero score identity, the Fisher information matrix, and the multivariate central limit theorem giveThe assumed regularity and a uniform law of large numbers giveConsequently, by the Slutsky theorem,
For the stated Gaussian shift model, direct expansion of the two normal distribution densities givesSince , this has exactly the limiting distribution above. Equivalently, both limits are normal with mean and variance . This is the local asymptotic normality of the model at .
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Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.Figure 3. Visual Studio Code extension installation.Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. - Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact





