how to transform non-normal data for downstream statistical analyses
Normality of dependent variable = normality of residuals?
Proving $\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$ I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$ I have some progress made, but I am stuck and could use some help. What I did: It holds that $$\sum\limits_{k=0}^{n}\cos(kx)=\sum\limits_{k=0}^{n}Re(\cos(kx))=\sum\limits_{k=0}^{n}Re(\cos(x)^{k})=Re(\sum\limits_{k=0}^{n}\cos(x)^{k})=Re\left(\cos(0)\cdot\frac{\cos(x)^{n}-1}{\cos(x)-1}\right)=Re\left(\frac{\cos(x)^{n}-1}{\cos(x)-1}\right) $$ For any $z_{1},z_{2}\in\mathbb{C}$ we have it that if $z_{1}=a+bi,z_{2}=c+di$ then $$\frac{z_{1}}{z_{2}}=\frac{z_{1}\overline{z2}}{|z_{2}|^{2}}=\frac{(a+bi)(c-di)}{|z_{2}|^{2}}=\frac{ac-bd+i(bc-ad)}{|z_{2}|^{2}}$$ hence $$Re\left(\frac{z_{1}}{z_{2}}\right)=\frac{Re(z_{1})Re(z_{2})-Im(z_{1})Im(z_{2})}{|z_{2}|^{2}}$$ Thus, $$Re\left(\frac{\cos(x)^{n}-1}{\cos(x)-1}\right)=\frac{(\cos(nx)-1)(\cos(x)-1)-\sin(nx)\sin(x)}{(\cos(x)-1)^{2}+\sin^{2}(x)}=\frac{\cos(nx)\cos(x)-\cos(nx)-\cos(x)+1-\sin(nx)\sin(x)}{\cos^{2}(x)-2\cos(x)+1+\sin^{2}(x)}=\frac{\cos(nx)\cos(x)-\cos(nx)-\cos(x)+1-\sin(nx)\sin(x)}{-2\cos(x)+2}=\frac{\cos(nx)\cos(x)-\cos(nx)-\cos(x)+1-\sin(nx)\sin(x)}{-2(\cos(x)-1)}= \frac{=\cos(nx)\cos(x)-\cos(nx)-\cos(x)+1-\sin(nx)\sin(x)}{-2(-2\cdot\sin^{2}(x/2))}=\frac{\cos(nx)\cos(x)-\cos(nx)-\cos(x)+1-\sin(nx)\sin(x)}{4\sin^{2}(x/2)}=\frac{\cos(nx)\cos(x)-\cos(nx)-\cos(x)+1-\sin(nx)\sin(x)}{4\sin^{2}(x/2)}=\frac{\cos(x(n+1))-\cos(nx)-\cos(x)+1}{4\sin^{2}(x/2)} $$ This is the part where I am stuck, I would appriciate any help or hint on how to continue. Edit: Given the corrections by André I get: $$(\cos(nx+x)-1)(\cos(x)-1)+\sin(nx+x)\sin(x)=\cos(nx+x)\cos(x)-\cos(nx)-\cos(x)+1+\sin(nx+x)\sin(x)$$ so $$\cos(nx+x)\cos(x)+\sin(nx+x)\sin(x)=\cos(xn+x-x)-\cos(nx)=0$$ Edit 2: I found anoter mistake in the above, I will try to correct Edit 3: When multiplying correctly the above it works out :-) Work the Spring There's a book that I read several years ago that I'd like to find.  However, I cannot remember enough about it to find it myself.  Neither the title nor author remain in my mind, nor any keyword or phrase that's unique enough for the Grand Google to nail for me.  Hence my desperate plea here.  In the book, there is an interstellar civilization composed of several power blocks.  The entire civilization did not fill the galaxy, but it did span several stellar systems.  Maybe 10s or 100s.  I seem to remember that one of the civilizations was a decentralized group referred to as the Brotherhood and they seemed to have been keen on preserving old learning and political ideals.  A derelict alien spacecraft from outside all of this had been discovered and at least two of the power blocks sent ships to recover it.  I don't recall if they knew at the time about the power the alien ship had, but they must have had some clue since there was some urgency in capturing this ship.  One of the ships sent to capture the Alien craft was commanded by somebody who was rather shaken up, to say the least.  He had had some big trauma in his past (like... the last ship he captained was annihilated with all hands.) so he was kinda bummed out about things and wasn't really putting his heart into getting this done.  He was assisted by the ship's AI who had a lot of soul-to-soul chats with the captain, and at one point almost relieved him of command because he was just not getting the job done.  I think the ship's name was Boaz.  These were the "good guys".  There was at least one other ship sent from the "bad guys."  The alien craft had some remarkable powers.  The interior volume was apparently infinite and the operators could see and manipulate anything in the universe.  There might have been some example given where all the rats somewhere were found and disposed of.  Whoever had control was tempted by the ole absolute power corrupts absolutely problem, but nobody but nobody could stop them because of their power.  They could eat and multiply within the ship and become real bad-asses.  But alas... (or hurray, depending upon which side you're one) everything fails eventually, and whoever controlled the ship would eventually succumb to some horrible end.  Whereupon the ship became derelict again until some other lucky race found it and repeated the cycle.  The ship had an AI that was real interested in taking the power for itself, but there was some mechanical, spring-loaded, mechanism outside of its control that needed to be worked in order to execute power.  The AI apparently never figured out how to "work the spring" and this made him crazy angry.  The ending is equally foggy for me.  I'm pretty sure the good guys got the ship.  They agonized over what to do with it.  Apparently it was indestructable and nobody wanted to taste the temptation of all that power.  In the end they dumped it on the surface of a neutron star.  There it sat forever more, probably with the crazy AI really pissed off about being there.  If you've read this book, you'll recognize it immediately from these clues.  I'm looking forward to finding out the name of this book.  Thanks! I want to apply a macro to all the cells of a column with the cell content as the input to the macro. Is this possible?  For example if the cell contents have the word TEST, the macro should take TEST as its argument. How would one format an Excel sheet so numeric fields are set to numeric field types when importing to ArcGIS 10.2? Entropy of sum is sum of entropies Show that a Cauchy sequence has a fast-Cauchy subsequence How to select lenses for outdoor sports photography? I would like to get into sports photography, and would probably want a better telephoto lens so I can get better close up shots.  What other factors should I keep in mind as I make my choice? Does IS (or VR) matter?  What is a good focal length?  I'll probably be shooting mostly little league baseball/softball, church softball leagues, and kids soccer. There are similar questions for  and , but I'm after a Windows solution here.  The problem is as follows: I want to write several (js) script files in a folder, and have a  program monitor that folder for file changes and new files being added, and run a command whenever that happens (to compile them all into one single file).  The solution has to:   Monitor both file changes and new files being added, in a folder. Run a command only if there is any change.   It would be best if it either is a built-in solution (like a JScript or VBscript snippet), or something that does not require installation. I'm trying to find  $$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$   I tried couple of methods: Stolz, Squeeze, D'Alambert  Thanks!  Edit: I can't use Stirling. Prove or disprove that $f(x) = \sin x/x$ is uniformly continuous over the interval $(0,1)$? I am thinking of $\lim_{x \to 0} \ {f(x)} = 1$ but I am confused as $x=0$ is not in the domain and also I want to write an $\epsilon$-$\delta$ proof. So any help is much appreciated!  Thanks! In natural language parsing, what is the feature function? I'm working in the area of natural language processing, to be specific I'm reviewing the parsers that take advantage of data-mining techniques.   I've read an introduction to natural language processing, specifically the  by Della Pietra and others. In that paper, the authors explain that features are indicator functions that take into account certain aspects of the text. The problem is that this aspects affect the maximum entropy function.  In the case of parsing, I have problems understanding the concept of features. For parsing one must have:   Two sets, $X$ for the sentences and $Y$ for the parses For each sentence $x \in X$, one must have the N-best list of candidate parses; denoted $GEN(x) \subset Y$. Training samples $(x_j,y_j)$. The $x_j$ are sentences and the $y_j$ are parses. And a feature mapping $\Phi(X,Y)$ that maps each pair (x,y) to a vector of feature values.   The problem of parsing will consist of finding the weight vector such that the following expression is minimized  $$ -\sum log P(y_j|x_j,w) + \lambda ||w||_1 $$  where   $$ P(y_j|x_j,w) = \frac{exp \{w^T \Phi(x_j,y_j)\}   }{ \sum_{u \in GEN(x)} exp\{ w^T\Phi(x_j,u) \}  } $$  My question is, what are the features in this case? Some codified version of the parsing tree? Are the features a statistical concept?  I will also appreciate any bibliography that can be useful in tackling this issue.  UPDATE: I've talked to a professor and he had just told me that the features are obtained after doing a classification algorithm based on the parse-tree. Do you know what kind of classification is done with the parse tree? When compiling a tex file including references (by Texstudio editor and Miktex distribution) using biblatex with bibtex backend and ieee style, the results of the displayed references in the output pdf is messed up at the "volume" and "issue number" entries as follows:   The volume label "vol." is missing (Ref. 1 &amp; 2 in the given example below). The numeric issue number "3" is missing (Ref. 1 in the given example). The issue number label "no." is displayed although NO input entry was being given (Ref. 2 in the given example).   If it helps, this behavior was noticed after the recent update of Miktex on March 11, 2017, including  package (ver. 1.2a 2017-03-02). As before which, everything was going fine!  So is this issue noticed by someone else? Is there any idea about the reason of such strange act by biblatex?  \documentclass{article} \usepackage{filecontents} \begin{filecontents}{reference.bib}     @article{author1,         author  = "George Green",         title   = "Basic latex",         year    = "2000",         journal = "Sharelatex Journal",         volume  = "10",         number  = "3",         pages   = "100--101"     }     @article{author2,         author  = "Carol White",         title   = "Basic studio",         year    = "2010",         journal = "Club Journal",         volume  = "20",         pages   = "200--201"     } \end{filecontents} \usepackage[backend=bibtex, style=ieee]{biblatex}  \addbibresource{reference.bib} \begin{document}     Hi there! \\     Ref 1 Green \cite{author1}. \\     Ref 2 White \cite{author2}.     \printbibliography  \end{document}   The output is as follows: If $n\times n$ matrix $A$ has eigenvalues $1,-1$ and $n\times n$ matrix $B$ also has eigenvalues $1,-1$, can I then say something about eigenvalues of $AB$ and $BA$? ML estimate of exponential distribution (with censored data) In Survival Analysis, you assume the survival time of a r.v. $X_i$ to be exponentially distributed. Considering now that I have $x_1,\dots,x_n$ "outcomes" of i.i.d r.v.'s $X_i$. Only some proportion of these outcomes are in fact "fully realized", i.e. the remaining observations are still "alive".   If I wanted to perform a ML estimate for the rate parameter $\lambda$ of the distribution, how can I utilize the non-realized observations in a coherent/appropriate manner? I believe they still contain useful information for the estimation.   Could someone guide me to literature on this topic? I am sure it exists. I am however having trouble finding good keywords/search terms for the topic. Is it acceptable to use "whose" when referring to an object? Whose is used for people (e.g. "Whose that girl?"), but is it acceptable to use whose for objects too? Is there another word I should use for objects?  To make the question clearer, when I ask "Whose that car?" I am expecting an answer similar to "It's X's car." where X refers to a person. I could say "a man whose opinion I respect." and whose opinion would refer to the opinion of a person. Could I say "the desk whose drawer was broken by my brother"? Should I replace whose with another word? How many passes do you need to wipe/shred your files to make them not undeletable? If you could travel to the center of the Earth (or any planet), would you be weightless there? Where $F$,$K$ are two fields. What does the element in the composite field $FK$ look like? All the elements are generated by the elements of $F$ and $K$? (combination of the elements of $F$ and $K$) I think $FK$ is pretty close to the free product of $F$ and $K$; is $FK \subsetneq F*K$? Thank you! Nature of gravity: gravitons, curvature of space-time or both? General relativity tells us that what we perceive as gravity is curvature of space-time.   On the other hand (as I understand it) gravity can be understood as a force between objects which are exchanging (hypothetical) virtual particles called gravitons, similar to the way electromagnetic forces are due to objects exchanging virtual photons?  At least at first glance, the two concepts seem mutually exclusive. Is there a description of gravity which includes both, or is this contradiction one of the problems in combining GR with quantum mechanics? How iptables tables and chains are traversed I know linux has 3 built-in tables and each of them has its own chains as follow:  FILTER: PREROUTING, FORWARD, POSTROUTING  NAT: PREROUTING, INPUT, OUTPUT, POSTROUTING  MANGLE: PREROUTING, INPUT, FORWARD, OUTPUT, POSTROUTING  But I can't understand how they are traversed, in which order, if there is. For example, how are they traversed when:   I send a packet to a pc in my same local network when I send a packet to a pc in a different network when a gateway receives a packet and it has to forward it when I receive a packet destinated to me any other case (if any)