Regarding multiple comparisons, could someone please explain me why the power of false discovery rate (FDR) is greater than the power of family-wise error rate (FWER)?
I have read that controlling FDR is less stringent than controlling FWER, such as in :     FDR controlling procedures exert a less stringent control over false discovery compared to familywise error rate (FWER) procedures (such as the Bonferroni correction). This increases power at the cost of increasing the rate of type I errors, i.e., rejecting the null hypothesis of no effect when it should be accepted.   But I was wondering how it is shown to be true mathematically?  Is there some relation between FDR and FWER?
I have downloaded Python 3.3 from the official site but no idea how to install it.   I'm using Ubuntu 12.04 Eeeeek - what happened to my @ salutation? Why was William Cage sent to the base? I have a  beautiful  inequality, but I can only prove part of cases. Given any positive real numbers $a,b,c$, we have   $$(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$$  How can we prove this statement? Do Wine Viruses only work while Wine is running? I'm looking for an algorithm that sorts strings similar to the way files (and folders) are sorted in Windows Explorer. It seems that numeric values in strings are taken into account when sorted which results in something like  name 1, name 2, name 10   instead of  name 1, name 10, name 2   which you get with a regular string comparison.  I was about to start writing this myself but wanted to check if anyone had done this before and was willing to share some code or insights. The way I would approach this would be to add leading zeros to the numeric values in the name before comparing them. This would result in something like  name 00001, name 00010, name 00002   which when sorted with a regular string sort would give me the correct result.  Any ideas? Choosing between $z$-test and $t$-test Background: I'm giving a presentation to colleagues at work on hypothesis testing, and understand most of it fine but there's one aspect that I'm tying myself up in knots trying to understand as well as explain it to others.  This is what I think I know (please correct if wrong!)   Statistics that would be normal if variance was known, follow a $t$-distribution if the variance is unknown CLT (Central Limit Theorem): The sampling distribution of the sample mean is approximately normal for sufficiently large $n$ (could be $30$, could be up to $300$ for highly skewed distributions) The $t$-distribution can be considered Normal for degrees of freedom $&gt; 30$   You use the $z$-test if:   Population normal and variance known (for any sample size) Population normal, variance unknown and $n&gt;30$ (due to CLT) Population binomial, $np&gt;10$, $nq&gt;10$   You use the $t$-test if:   Population normal, variance unknown and $n&lt;30$ No knowledge about population or variance and $n&lt;30$, but sample data looks normal / passes tests etc so population can be assumed normal   So I'm left with:    For samples $&gt;30$ and $&lt;\approx 300$(?), no knowledge about population and variance known / unknown.   So my questions are:   At what sample size can you assume (where no knowledge about population distribution or variance) that the sampling distribution of the mean is normal (i.e. CLT has kicked in) when the sampling distribution looks non-normal? I know that some distributions need $n&gt;300$, but some resources seem to say use the $z$-test whenever $n&gt;30$... For the cases I'm unsure about, I presume I look at the data for normality. Now, if the sample data does looks normal do I use the $z$-test (since assume population normal, and since $n&gt;30$)? What about where the sample data for cases I'm uncertain about don't look normal? Are there any circumstances where you'd still use a $t$-test or $z$-test or do you always look to transform / use non-parametric tests? I know that, due to CLT, at some value of $n$ the sampling distribution of the mean will approximate to normal but the sample data won't tell me what that value of $n$ is; the sample data could be non-normal whilst the sample mean follows a normal / $t$.  Are there cases where you'd be transforming / using a non-parametric test when in fact the sampling distribution of the mean was normal / $t$ but you couldn't tell? In an extract I am dealing with, I have 2 datetime columns. One column stores the dates and another the times as shown.  How can I query the table to combine these two fields into 1 column of type datetime?  Dates  2009-03-12 00:00:00.000 2009-03-26 00:00:00.000 2009-03-26 00:00:00.000   Times  1899-12-30 12:30:00.000 1899-12-30 10:00:00.000 1899-12-30 10:00:00.000 How do I protect Python code from being read by users? I am developing a piece of software in Python that will be distributed to my employer's customers. My employer wants to limit the usage of the software with a time restricted license file. If we distribute the .py files or even .pyc files it will be easy to (decompile and) remove the code that checks the license file. Another aspect is that my employer does not want the code to be read by our customers, fearing that the code may be stolen or at least the &quot;novel ideas&quot;. Is there a good way to handle this problem? How do I solve the partial differential equaiton  $u_{t} = (u_{x})^{-1}$  subject to the conditions $u(t,x) \geq 0$ and $u(0,0) = $0. Use the separation Ansatz $u(t,x) = f(t)g(x)$.  The inverse in confusing me Menu select item stuck on screen after context or command menu has closed Periodically when I select a menu command, the label or selected option gets "stuck" on screen and won't go away. I can close all open applications, including whichever one I was using when it got stuck, but it still won't go away.   In the screenshot below, I opened an new instance of IE just to show how the label stays on top. The label was not created by this instance of IE.    (I added the circle in MS Paint. That part isn't stuck.)  The label that gets stuck is the first menu command I select in IE. If a label is already stuck, a new one does not get stuck (regardless of which instance(s) of IE are involved). Based on this knowledge, I now just open IE on my secondary monitor, carefully open the context menu so the Properties command is in the bottom corner, and click it. This is not a solution.  The floating label never moves and is transparent to mouse input (if I click it, it's as if I clicked the item behind it).  The label does not go away if I close all running applications. I haven't tried stopping services or closing system tray items like Live Mesh.  The label does go away if I change the screen resolution and then change it back.  Any ideas how I can stop this from happening? It's happened a half dozen times and it's becoming quite disrupting to my work. Why Didn't People Think That Voldemort Died After His Attack on Baby Harry? After the Avada Kedavra curse rebounded from Harry onto Voldemort, Voldemort disappeared. I'm curious as to why so many key players believed Voldemort was still "alive" somewhere, his powers decimated, but, under the right circumstances, able to return to full power. It seems the witches and wizards in chapter one of Philosopher's Stone, The Boy Who Lived, were happy to believe that Voldemort was gone forever. Why didn't everyone believe Voldemort had died, rather than just disappeared? Was it the lack of a body? Or were there other reasons?  Here's a couple of quotes from Philosopher's Stone if anyone wants to review them. Otherwise, just disregard.     ‘A fine thing it would be if, on the very day You- Know-Who seems to have disappeared at last, the Muggles found out about us all. I suppose he really has gone, Dumbledore?’   ‘It certainly seems so,’ said Dumbledore. ‘We have much to be thankful for.’   Philosopher's Stone - page 13 - Bloomsbury - chapter 1, The Boy Who Lived   ‘But what happened to Vol– sorry – I mean, You-Know-Who?’   ‘Good question, Harry. Disappeared. Vanished. Same night he tried ter kill you. Makes yeh even more famous. That’s the biggest myst’ry, see ... he was gettin’ more an’ more powerful – why’d he go?   ‘Some say he died. Codswallop, in my opinion. Dunno if he had enough human left in him to die. Some say he’s still out there, bidin’ his time, like, but I don’ believe it. People who was on his side came back ter ours. Some of ’em came outta kinda trances. Don’ reckon they could’ve done if he was comin’ back.   ‘Most of us reckon he’s still out there somewhere but lost his powers. Too weak to carry on. ’Cause somethin’ about you finished him, Harry. There was somethin’ goin’ on that night he   hadn’t counted on – I dunno what it was, no one does – but somethin’ about you stumped him, all right.’   Philosopher's Stone - pages 46-47 - Bloomsbury - chapter 4, The Keeper of the Keys Evaluate $$\int \frac{2-3x}{2+3x}\sqrt\frac{1+x}{1-x}d x$$ What substitution should I use ? $\sqrt\frac{1+x}{1-x}$ suggests $x=cos2\theta $ but its not useful in $\frac{2-3x}{2+3x}$ Where do the bipolar jets of black holes come from? How are they formed? And why are they so bright? Does Qt have a C interface? How to install an application that requires gksu package on ubuntu 18.04? I downloaded the Upwork desktop App on ubuntu 18.04. When I open it with Gdebi package installer I saw this error:    I tried to install gksu manually via sudo apt install gksu but I got this error  Building dependency tree        Reading state information... Done Package gksu is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source  E: Package 'gksu' has no installation candidate   How can I install this software on ubuntu 18.04 (I download it yesturday so it is the most recent version on upwork website) ? Sci-fi story where all 'citizens' had shares in a global company Not everyone was a citizen though, there was a significant underclass who had no shares.  One character was a celebrity in some sort of combat game who was followed by camera drones &amp; used drugs to suppress puberty so he waa an eternal 'child', I have the impression that medical science allowed virtual immortality short of accidents.  One character (may have been the same one) had had a 'previous life' &amp; had committed personality suicide by some form of mind wipe bequeathing his worldly possessions to the new personality that would eventually develop in his old body.  I think the motive for this 'suicide' was essentially boredom, he'd lived a long time.  The mind wipe in question was pretty extensive &amp; all encompassing, he had to be cared for &amp; educated from scratch as if a newborn baby in an adult body, nappies &amp; all (but I'm pretty sure this was purely backstory in the book I read).  He also had miniature nano organisms (or bots?) like miniature worms or millipedes that complimented his immune system &amp; aided healing, they were self replicating, pretty sure they were some sort of bioengineered organism tailored to his DNA rather than bots  but not 100% on that.  There was an organism spanning a large area on the planet that absorbed (or consumed) anything that wandered into it but saved the DNA of anything it had previously absorbed &amp; might randomly reproduce any animal plant or individual it had previously absorbed fully formed.  One of the characters fell into this but got pulled out before the full effects took hold, as a result he grew leaves from his head &amp; his immune system was then constantly battling with the 'infection'.  When did I read this? probably some time in the 80's in the UK.  When was it written? I haven't a clue, sorry.  It was a paperback, probably something I had out of a library, because I never get rid of anything I've bought myself so I'd still have it &amp; be able to dig it out if it wasn't a library loan.  I have lots of fragments of world background &amp; 'colour' that seem to have stuck in my mind there, but a complete lack of any story arc to go with them. RSA, DES, AES, etc., all use (relatively) complicated mathematics to encrypt some message with some key. For each of these methods, there have been several documented vulnerabilities found over the years. Some even believe that there is a vulnerability in AES known only to the NSA.  On the other hand, the one-time pad is not only ridiculously simple, but also has been shown to be impossible to crack (if used appropriately).  Say you have some message to be encrypted with length n. Simply generate a random string with length m where m is at least as large as n, and perform modular addition of the random string onto the message.  As long as the random string is generated with a high-enough-quality random number generator, and as long as the same one-time pad isn't reused, it should be impossible to crack.  If this is true, all we would need to perfect are fast, cryptographically secure deterministic random number generators. Use the key as a starting seed for the RNG and you have a powerful uncrackable encryption scheme which is simple to understand and implement.  Blum-Blum-Shub is a fast, cryptographically secure PRNG, and there are others too.  I have written a C++ program which uses two 2048-bit primes (which match the necessary criteria) for Blum-Blum-Shub and uses a password which the user enters - converted into a starting seed - to run byte-wise modular addition. The process is fast, deterministic, and if the literature I have read is correct then it should be very secure.  So why is so much time and effort invested in coming up with convoluted encryption cyphers when mathematics shows that the simplest is often the best? Can't install jdk on Fedora with yum nor with rpm Express solutions of equation $ \tan x= x $ in closed form I know that the equation $$ \tan(x) = x $$ can be solved using numerical methods, but I’m looking for a closed form of the solutions. In my opinion, having only numerical solutions means that we don’t know the problem, and sooner or later, we’ll be able to find a closed-form solution or at least a power-series solution. I’m looking for an explicit form of a sequence $ (x_{n})_{n \in \mathbb{Z}} $ in $ \mathbb{R} $ such that  $ 0 &lt; x_{0} &lt; \dfrac{\pi}{2} $, $ \tan(x_{n}) = x_{n} $ for all $ n \in \mathbb{Z} $, and $ (2 n - 1) \dfrac{\pi}{2} &lt; x_{n} &lt; (2 n + 1) \dfrac{\pi}{2} $ for all $ n \in \mathbb{Z} \setminus \{ 0 \} $.  The existence of such a sequence follows from the continuity of $ x \mapsto \tan(x) - x $ over its domain. Its uniqueness follows from the monotonicity of $ \tan(x) - x $ over the intervals $$ \left( (2 n - 1) \frac{\pi}{2},(2 n + 1) \frac{\pi}{2} \right), \quad n \in \mathbb{Z}. $$