Assume finite second moments and set . By linearity of expectation, centring gives . Expanding its square and using the definitions of variance and covariance,
No independence assumption is required for this variance of a sum identity.
Write . Equality of -values defines an equivalence relation induced by a function: equality is reflexive, symmetric, and transitive, so all three properties hold for . The equivalence class of is
For this is the circle centred at the origin with radius ; the remaining equivalence class is the singleton .
By the generalized pigeonhole principle, 13 cards guarantee cards in one suit. The guarantee is sharp: 12 cards can consist of three cards from each suit, with no four cards of a common suit.
Set , using the ceiling function. If no box contains at least objects, all boxes contain at most , so the total number of objects is at most . Taking the contrapositive, a total greater than forces some box to contain at least objects. But , hence . This proves the generalized pigeonhole principle: some box contains at least objects.
Let contain the multiples of among . Its cardinality is , using the floor function. Because are pairwise coprime, their pairwise intersections correspond to divisibility by , and their triple intersection to divisibility by . The inclusion-exclusion principle gives
Each of the elements is either included or excluded, so the rule of product gives . Alternatively, partition the power set into disjoint families according to cardinality. The family of -element subsets has size equal to the binomial coefficient . The rule of sum therefore gives
The rule of product says that a sequence of choices, with possibilities at stage regardless of earlier choices, has possible outcomes. Applied to the Cartesian product of finite sets, it gives . Here independence means that the available choices do not restrict one another; no probability assumption is needed.

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!
We have two killer features:
  1. 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
  2. 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.
    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.
  3. https://raw.githubusercontent.com/ourbigbook/ourbigbook-media/master/feature/x/hilbert-space-arrow.png
  4. Infinitely deep tables of contents:
    Figure 6.
    Dynamic article tree with infinitely deep table of contents
    .
    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