For an estimator of , its bias and mean squared error are
When the second moment exists, expanding around gives the bias-variance decomposition .
For the exponential distribution with rate , the change of variables gives
The integrals are one and two by integration by parts. Hence .
By part iii, is an integer with probability zero. Thus almost surely, and
The pointwise inequalities imply
Putting and rearranging yields
This extracts bounds for an infinite series from simple rounding inequalities.
For a nonnegative integer-valued random variable , the tail-sum formula states . It follows by writing and using nonnegativity to interchange sum and expectation.
With , this gives
The Riemann zeta function is finite here because .
For an integer , is the same event as . Integration for the Pareto distribution gives
The second identity subtracts two successive tail probabilities, equivalently integrating over . If includes zero, the extra case is and , since almost surely.
The condition guarantees a finite first moment. Direct integration gives
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.
Put . Then
Uniqueness of the probability generating function therefore gives
This is Bernoulli thinning of a geometric count, using the convention of counting failures before success.
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 .
Here , so
This identity holds throughout , and its power-series extension has radius .
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 .
Differentiate the probability generating function inside its radius of convergence:
Since these are and , respectively,
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.
The equilibrium of the first-order linear recurrence is . Subtracting it gives
Consequently
For , interpret the formula at through the initial condition: , while for . For every , .
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.
Only upward and downward moves change the value, so the conditional expectation is
Thus .
Conditional on , urn 1 has red and blue balls, while urn 2 has red and blue balls. The independent selections therefore give
Every 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.

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