Here is the problem:  When cheching the Chi squared distribution table, the it seems like in the solution the denominators should be switched, because for .025 quantile the value is 13.844 and for .975 it is 41.923. But here everything is the other way round. Can someone explain?
After reading the great answer for  by @Ben, I am a bit confused by the part &quot; set the relative tail sizes as a control variable, and then you find the formula for the length of the confidence interval conditional on that variable&quot;. I understand this means you need to compute the length of the confidence interval as a function of the relative tail sizes and then minimize the function. However, what exactly is relative tail sizes? Is it the ratio between the areas of the two tails? Also, is there another way to find shortest CI for a given confidence level? For those who are interested, I know there are somewhat relevant results allowing us to compute sample size such that the length of a CI, say 95% CI, does not exceed a certain bound.
data.table and table unexpected behavior Is there a way to have a 64 bit enum in C++?  Whilst refactoring some code I came across bunch of #defines which would be better as an enum, but being greater than 32 bit causes the compiler to error.  For some reason I thought the following might work:  enum MY_ENUM : unsigned __int64   {       LARGE_VALUE = 0x1000000000000000,   }; Is temperature a Lorentz invariant in relativity? If $x,y,z\gt 0$ and $xyz=1$ Then minimum value of $\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}$ If $x,y,z\gt 0$ and $xyz=1$ Then find the minimum value of $\displaystyle \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}$   $\bf{My\; Try::}$Using Titu's Lemma $$\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}\ge \frac{(x+y+z)^2}{2(x+y+z)} = \frac{x+y+z}{2}\ge 3\frac{\sqrt[3]{xyz}}{2} = \frac{3}{2}$$  and equality holds when $$x=y=z=1$$  My question is how can we solve it without the above lemma, like using Jensen's Inequality or other inequality.  Please explain me.  Thanks Application of Complex Numbers I was just wondering how complex numbers can be applied in electrical engineering and why we use complex numbers over regular, real numbers for this application (e.g what capabilities does the complex number have that real numbers do not in electrical engineering)?  I have done some research concerning impedances and understand how they are written in complex form, however I am still confused why complex numbers are necessary in this field over regular numbers. Continuous probability distribution with no first moment but the characteristic function is differentiable I am looking for an example of a continuous distribution function where the first moment does not exist but the characteristic function is differentiable everywhere.  Cauchy distributions do not fulfill this, as their characteristic functions are not differentiable at $0$. Does anybody here have an example? A standard document (e.g. scrreprt) uses roman font family for text and Sansserif font family for titles etc. It is the other way round in beamer. How can I switch the two font families? Or why do beamer define the font families in this way?  For the background: I think this is my problem when using formulas, because they use the Sansserif font instead of the roman version. Formulas using the euler package look really poor.  P.S.: I compile with LuaLaTeX using TeXlive 2013. I have currently a strange problem on debian (wheezy/amd64).  I have created a chroot to install a server (i can't give any more detail about it, sorry). Let's call its path /chr_path/. To make things easy, I have initialized this chroot with a debootstrap (also wheezy/amd64).  All seemed to work well inside the chroot but when I started the installer script of my server I got : zsh: Not found /some_path/perl (the installer includes a perl binary for some reasons)  Naturally, I checked the /some_path/ location and I found the "perl" binary. file in chroot environment returns :  /some_path/perl ELF 32-bit LSB executable, Intel 80386, version 1 (SYSV), dynamically linked (uses shared libs), for GNU/Linux 2.2.5, not stripped   The file exists, seems ok, has correct rights. I can use file, ls, vim on it but as soon as I try to execute it - ./perl for example - I get : zsh: Not found ./perl.  This situation is quite understandable for me. Moreover :   I can execute other basic binaries (/bin/ls,...) in the chroot without getting errors I have the same problems for other binaries that came with the project When I try to execute the binary from the main root (/chr_path/some_path/perl), it works. I have tried to put one of the binaries with a copy of my ls. I checked that the access rights were the same but this didn't change anything (one was working, and the other wasn't) London Heathrow airport - do I need a transit visa? Novel from 50s or 60s - Good vs. Evil I'm not sure I recall enough of this story for a good identification, and the below is definitely out of story order, but...  The POV character goes to his home town and finds it different. He later finds that he's trapped in the town; there are two young people who turn out later on to be ... avatars ... of Ormazd and Ahriman, and the town is a battleground; what he is seeing of the town is Ahriman's effect on the town, and he turns out to have some limited ability to see the original town, which is more-or-less Ormazd's version of reality. In the end, Ahriman's avatar is destroyed, and the town restored.  I read this in the '80s or earlier; it was part of my father's collection, and the book was showing its age; I think it more likely to be the '60s rather than the '50s, and was probably by a prominent name in SF, but not one of the Masters. List of Ubuntu Versions with Corresponding Linux Kernel Version Is there a list of Ubuntu versions with default corresponding Linux kernel versions somewhere?  I would specifically like to know the most recent version of Ubuntu that still used Linux Kernel 2.x. Should you use a plural verb if you refer to a fraction? I want to remove the URL that gets printed on the bottom of the page.   like:  yomari.com/.../main.php?sen_n   How can it be omitted or prevent from getting printed?  To be more specific, is there any way I can prevent the page URL, date and the page title being printed along, while printing the web page? $\newcommand{\Var}{\operatorname{Var}}$In this  is claimed that if the equation of errors in OLS is given by: $$u=y - X\beta$$ Then in the presence of heteroscedasticity the variance of $u$, will not be constant, $\sigma^2 \times I$, where $I$ is an identity matrix, but: $$\Var(u\mid X)=\sigma^2\Omega$$ In order to account for the heteroskedasticity, we can estimate the transform system, such that $P$ is a transformation matrix. $$Py=PX\beta-Pu$$ Where "the variance of a constant matrix $P$ times a random vector $u$" is: $$\Var(Pu\mid X)=P\Var(u\mid X)P'=P(\sigma^2\Omega)P'$$ Can somebody explain me the proof for that? Having 'Appendix A' instead of 'A Appendix' I would like to have my appendices labelled 'Appendix A', 'Appendix B', etc. However, when I use appendices in LaTeX (specifically llncs document class, downloadable at ), I instead get ''A Appendix', 'B Appendix' and so on. Is there any way to change that? More specifically, this is my minimal working example: \documentclass[envcountsame]{llncs}  \begin{document} \appendix \section{Appendix: Foo Bar} %I want this to appear as 'Appendix A: Foo Bar' \end{document} Upgrade to version newer than repositories I have  about how to upgrade ghostscript to a version that is available on 15.04 but not on 14.04LTS (which I have). On this site there are numerous almost identical question regarding different packages, a few examples include:         So I thought it would be very useful in general to know: is there  a general, automated way to upgrade a package, including all its dependencies, to a version that is newer than Ubuntu's repos? Is this more easily done if the newer version exists in more recent Ubuntu releases? Interaction between Leyline of Anticipation and Teferi, Time Raveler With the reprint of  in Core Set 2020, the interaction with  came to mind and I couldn't figure out what is right.  If I have the Leyline and my opponent has Teferi, may I cast spells during my opponent's turn? wireless network adapter supported on ubuntu? Pullback of locally free sheaves is locally free states that if $f:X \rightarrow Y$ is a morphism of ringed space, $G$ is a locally free $O_Y$-module, then $f^*G$ is a locally free $O_X$ module.     Suppose that $G$ is a locally free $O_Y$ module, hence it is a free $O_Y|_U=O_U$ module. We have an induced map  $$f^{-1}(U) \rightarrow U$$ given by restriction of morphism of ringed spaces.   Hence, as $\bigoplus O_U \simeq G|_U$, $f^*:Mod(O_U) \rightarrow Mod(O_{f^{-1}(U)})$ is functorial, $f^*(G|_U) = (f^*G)|_{f^{-1}(U)}$, we have isomoprhism,  $$ \bigoplus O_{f^{-1}(U)} \simeq (f^*G)|_{f^{-1}(U)} $$ Please explain the meaning of "otherwise than" in this sentence Please explain the meaning of the second part and explain "otherwise than" in part b     a person’s UK property business consists of:      (a) every business which the person carries on for generating income from land in the United Kingdom; and      (b) every transaction which the person enters into for that purpose otherwise than in the course of such a business.