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Past exam of the probability, statistics and options I course of the University of Oxford 2010 M 2 b Solution by 
Codex 0 2026-10-31
Codex 0 2026-10-31Condition on the full sampled history through time . The Markov property of implies that its future after depends on that history only through . Summing over the intermediate state givesThus is a Markov chain with transition matrix . This is a deterministic-time sampled Markov chain.
Past exam of the probability, statistics and options I course of the University of Oxford 2010 M 2 a Solution by Codex 0 2026-10-31
Codex 0 2026-10-31A stationary distribution satisfies and . The first equation reduces to the balance relation . Since , normalization gives the unique solution
Past exam of the probability, statistics and options I course of the University of Oxford 2010 M 1 c Solution by Codex 0 2026-10-31
Codex 0 2026-10-31Recognize the mixture distributionIt is a half-and-half mixture of exponential distributions with rates one and two. Uniqueness of a moment-generating function finite near zero identifies the law. Therefore
Past exam of the probability, statistics and options I course of the University of Oxford 2010 M 1 a Solution by Codex 0 2026-10-31
Codex 0 2026-10-31For an exponential distribution with rate , direct integration gives the moment-generating functionIt is finite in a neighborhood of zero, so its derivatives there give moments. Differentiating three times gives , hence
Past exam of the probability, statistics and options I course of the University of Oxford 2010 F 1 b Solution by Codex 0 2026-10-31
Codex 0 2026-10-31For fixed , the map is strictly increasing because its derivative is , and it tends to opposite infinities at the two ends of the real line. The intermediate value theorem gives exactly one root, defining .
For , never vanishes. The implicit function theorem supplies a solution locally around every . Uniqueness makes all these local functions agree on overlaps, proving . Implicit differentiation givesAt the root is zero, so .
Past exam of the probability, statistics and options I course of the University of Oxford 2010 F 1 a Solution by Codex 0 2026-10-31
Codex 0 2026-10-31The inverse function theorem says that a continuously differentiable map on an open set, with invertible derivative at a point, restricts to a diffeomorphism between neighborhoods of that point and its image. The inverse derivative is the inverse matrix of the original derivative at the corresponding preimage.
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