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. It has nondecreasing right-continuous integer-valued paths with unit jumps almost surely. The independent increment requirement distinguishes the process from merely having Poisson marginals.

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / M / M4 / a / Solution by

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The moment generating function is

M_{X} (θ) = E [e^{θX}]

for real

θ

where this expectation is finite. Existence on an open interval about zero determines the distribution and permits differentiation there to obtain moments, when the usual local integrability conditions hold.

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / F / F2 / b / Solution by

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Define the scalar function

H (s) = ⟨ f (x + s (y - x)), z ⟩

[0, 1]

. It is continuous there and differentiable inside, with derivative

⟨ df (x + s (y - x)) (y - x), z ⟩

by the chain rule. The one-dimensional mean value theorem gives

H (1) - H (0) = H^{'} (θ)

for some

0 < θ < 1

, proving the displayed vector-paired version. No claim of a single intermediate point giving a vector equality without pairing is required.

For the final assertion, use the scalar mean value theorem directly on

s \mapsto g (s h)

: it is continuous at

s = 0

and differentiable for

s > 0

. Thus

g (h) - g (0) = \nabla g (θ_{h} h) \cdot h

(1)

for some

0 < θ_{h} < 1

. The assumed limits of all partial derivatives imply

∥\nabla g (x) ∥ \to 0

x \to 0

x \neq = 0

. Consequently

∣ g (h) - g (0) ∣/∥ h ∥ \leq ∥\nabla g (θ_{h} h) ∥ \to 0

. Hence

d g (0) = 0

and

g

is differentiable there. This argument uses only differentiability on the interior of the line segment, so it does not assume the conclusion at its endpoint.

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / F / F2 / a / Solution by

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The

i

th partial derivative is the limit

\partial_{i} f (x_{0}) = lim_{t \to 0} [f (x_{0} + t e_{i}) - f (x_{0})] / t

. For a nonzero vector

v

, use the directional convention

\partial_{v} f (x_{0}) = lim_{t \to 0} [f (x_{0} + t v) - f (x_{0})] / t

, without normalizing

v

. Differentiability means

f (x_{0} + h) = f (x_{0}) + L h + o (∥ h ∥)

for a linear map

L = df (x_{0})

For the given example take

v = (v_{1}, v_{2})

. If

v_{1} \neq = 0

, then

f (t v) = t^{2} v_{2}^{3} / v_{1}

, so its directional difference quotient tends to zero. If

v_{1} = 0

, the function vanishes along that line. Thus

\partial_{v} f (0) = 0 for every v \neq = 0

. However along

(x, y) = (t^{3}, t)

with

t \neq = 0

, one has

f (x, y) = 1

, while

f (0, 0) = 0

. Hence the function is neither continuous nor differentiable at zero. Existence of all directional derivatives, even with a linear dependence on the direction, does not by itself imply total differentiability.

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / E / E4 / c / Solution by

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The difference map

(z, w) \mapsto z - w

is continuous and

X

is its inverse image of the open set

C ∖ {0}

, so

X

is open in

C^{2}

. For fixed

z

, the equivalence class is

{z} \times (C ∖ {z})

, homeomorphic to a punctured plane. It is path-connected: two points can be joined by a polygonal path avoiding the missing point. Thus all equivalence classes are connected.

The projection

π

is continuous and onto, since

(z, z + 1) \in X

. If

U

is open in

X

, it is open in

C^{2}

because

X

is open. Product-coordinate projection is open: around any

(z, w) \in U

there is a product of open disks contained in

U

, whose first disk is a neighborhood inside

π (U)

. Hence

π (U)

is open.

Its fibers are precisely the equivalence classes, so it induces a bijection

\overset{π}{ˉ} : X / R \to C

. Continuity follows from the quotient topology and

π = \overset{π}{ˉ} \circ q

. If

V

is open in

X / R

, then

q^{- 1} (V)

is open in

X

and

\overset{π}{ˉ} (V) = π (q^{- 1} (V))

is open. Thus the inverse is continuous too, proving

X / R ≅ C

as topological spaces. The plane is connected, so the connected-fiber equivalence in (b)(ii) gives

X is connected

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / E / E4 / b / ii by

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / E / E4 / b / i by

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / E / E4 / a / iii by

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / E / E4 / a / ii by

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / E / E4 / a / i by

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / E / E3 / b / v by

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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!

Video 1.

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-calculus
Articles 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/derivative
Video 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:
- 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
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.
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 4.
Visual Studio Code extension tree navigation
.
Figure 5.
Web editor
. 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.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
. Source.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade

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

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 Pinned article: Introduction to the OurBigBook Project