With the same separation of variables ansatz, gives
On , integration yields and . The choice , gives a polynomial that extends smoothly to all real :
It is nonconstant and real everywhere, and , including on the axes.
Use separation of variables with on a region where the factors do not vanish. The equation becomes , so
The separated equations are and , giving . Choosing gives the nonconstant real solution
Although the division used , the resulting solution extends smoothly across and directly satisfies everywhere.
Successive substitution in the matrix recurrence relation gives . In the matrix power of , every adjacent cancels, so . Consequently
Here for , and . The formula also gives correctly at .
Let and . The specified entries mean . Multiplication of determinants therefore gives
This is also the direct row- and column-scaling rule, and it includes zero scaling factors.
Multiplying an matrix by scales every row, giving a factor in its determinant. Also , since . Hence
Only needs to be invertible; the result remains valid when .
The first initial condition gives . The original first equation gives , whereas the general solution gives . Thus and
The corresponding has , checking the other initial condition.
The characteristic equation is , with roots . Therefore
If the second component is also wanted, the relation gives
These are damped oscillations associated with the complex conjugate eigenvalues of the coefficient matrix.
The first equation gives . Differentiate it and eliminate using the second equation:
Thus the required second-order linear differential equation is
Conversely, a solution of this equation together with solves both original equations, so the elimination loses no solutions.
The characteristic equation has a repeated root , suggesting the substitution . By the product rule,
so . The inhomogeneous linear differential equation reduces to . Two integrations give
Therefore
The two arbitrary constants account for the repeated-root homogeneous solution.
The characteristic equation of the homogeneous linear differential equation is
The real general solution is . The first boundary value gives . At , and , so . Hence
The probability density is an even function. Its derivative simplifies to
It increases for and decreases for , with maximum height . Thus the most probable value, or mode, is . The original sketch shows the symmetric, slightly flattened peak and long tails.
Figure 1.
Density of a standard Cauchy variable after an independent uniform displacement
. Details.
There is a genuine qualification to the request for a mean: the mean does not exist. The integral representation shows that as , so
The two divergent integrals cannot be subtracted to define an expected value. Symmetry does give the Cauchy principal value , but that is not the mean. Equivalently, adding the bounded variable cannot make the first absolute moment of a Cauchy distribution finite.
A uniform distribution has constant probability density on its interval, and the integral of that density must be one. The interval has length one, so
Changing either endpoint value has no effect on the probability distribution.
Use the law of total probability to average the conditional interval probability over :
The integrands are nonnegative, so Tonelli's theorem permits interchanging the integrals. Thus
Reading off the probability density gives the convolution of probability densities:
The independence of random variables means that fixing leaves the conditional probability density of equal to . Given that value of , is the translate . Hence
At a continuity point of , the expression is . Since , this conditioning uses a conditional probability density, rather than division by the probability of the single event .
Apply the preceding interval formula on a short interval and use the change of variables :
Comparison with the defining integral for a probability density gives
At continuity points, this can also be read as .
Subtracting the constant translates the event without changing its probability. Therefore
This is a translation of the interval in the probability distribution of .
Writing and comparing Cartesian coordinates gives
The unique solution, guaranteed by the basis property, is . Thus

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