$X$ is a non negative ramdom variable, prove that $EX = \int_{0}^{\infty} P(X&gt;t)dt$. Well I don't know how to start to prove this, I have a clue that says to me that is: write $P(X&gt;t)$ as the integral of the index function and use the Fubini's theorem.
Let $X$ be a non-negative random variable and $F_{X}$ the corresponding CDF. Show,   $$E(X) = \int_0^\infty (1-F_X (t)) \, dt$$   when $X$ has : a) a discrete distribution, b) a continuous distribution.   I assumed that for the case of a continuous distribution, since $F_X (t) = \mathbb{P}(X\leq t)$, then $1-F_X (t) = 1- \mathbb{P}(X\leq t) = \mathbb{P}(X&gt;  t)$. Although how useful integrating that is, I really have no idea.
I'm trying to prove that if $\alpha(t)=(x(t),y(t))$ is a $C^2$ regular curve $(\alpha'\neq0)$ with constant and positive curvature, then $\alpha$ is on the circunference and if $\alpha$ is the circunference, then its curvature is constant.  I solved the second part since $\alpha$ is the circunference, then $\alpha(t)=(r\cos\theta,r\sin\theta)$, so $|\alpha'(t)|=r$, I need help in the first part.  Thanks a lot. Why is Tony Stark in the Iron Man suit unaffected by G-forces, inertia and blunt force trauma? Tony's suit is an engineering marvel, and they explain many facets of the technology to help give legitimacy to the special powers, abilities and weapons it has.  I never read the Iron Man comics, but in the movie they really don't have a technological explanation for how he miraculously seems to be unaffected by crushing G-forces of sudden acceleration, the inertia of sudden stops and hair-pin turns, and especially the blunt force trauma from getting smashed in the face with Thor's hammer.  Certainly it appears the suit takes punishment, but Tony always remains unscatched, never a collapsed lung, broken bone or even a bloody nose.  In my infantile understanding of theoretical physics, the same technology that would be used to counter G-forces and sudden blunt force trauma to the body inside would be the very same technology used for  which in my honest opinion would be a much more stunning technological achievement than his heralded Arc Reactor.  Was their ever an explanation proferred for this and if so was it indeed attributed to Anti-Gravity? I am using Sql Server 2008 on Small Business Server 2003; client is using WinXP.   I added a user to my Active Directory Security Group; Why can't this user immediately access the database? It seems there is a delay before the user is recognized in Sql Server.   I am using AD Security Groups for permissions expressly so that I don't need to add individual users in Sql Server. So effectively I don't need to do anything but add the user to the AD Security Group in order to grant access.  But for some reason Sql Server doesn't immediately recognize the addition. I've seen this a number of times. I add the user to the group but that user can't access data until the next day. It seems that it doesn't query the Active Directory in real time. Can you confirm that is so?  What do I need to do so that Sql Server "refreshes" the list of users from Active Directory? Is a black hole singularity a single point? General relativity is expressed in terms of differential geometry, which allows you to do interesting things with the coordinates: multiple coordinates may refer to a single point, eg. the  has whole lines at the top and the bottom that correspond to the North and South poles, respectively; or a single coordinate may refer to multiple points, for example by using inverse geometry the origin refers to all the points infinitely far away.  So, is the singularity just a single point in the curved spacetime, or can it be a more extended object, described by a single coordinate? Two electrons, or an electron and a proton, interact with each other because of the Coulomb potential, which can also be seen in the Schrödinger equation (which is the equation that describes the motion for elementary particles).  But how do an electron and a photon interact? Why can a photon kick up an electron to a higher atomic orbital, or why is a photon emitted when an electron jumps down to a lower orbital? A photon do not have any charge, so the interaction cannot be caused by the Coulomb potential. So what is it that causes the interaction, and how does that affect the Schrödinger equation? (Or well, the relativistic version of the Schrödinger equation, since we're dealing with photons.) Bounding lines around tax nodes I am trying to draw lines around a number of nodes. I very much like the definition of \convexpath definition given , but it is not working well on my setup (MacTex 2014 with all available updates).   I tried to compile , but what I get is the following, which is clearly not what I should be getting. Any ideas would be much appreciated. I've seen a couple other questions with this error message but it seemed that the host and ip in question were ones the OP knew. These two are unrelated to my server.  Where and why is postfix generating this error?     Jul 24 17:42:14 email postfix/smtpd[32277]: warning: hostname kimtake.respectedopinion.com does not resolve to address 62.210.28.114   Jul 24 17:42:14 email postfix/smtpd[32277]: connect from unknown[62.210.28.114]   Jul 24 17:42:14 email postfix/smtpd[32277]: disconnect from unknown[62.210.28.114] When installing or booting from USB flash drive screen goes blank When I was installing Ubuntu 11.10 64-bit on my new PC from a flash drive since the only computers I have do not have disk drives, I ran into a problem.  After the ASUS logo when switching on the PC, and pressing F8 to select the USB flash drive, there was the options thing as normal that allows me to select to either boot form the flash drive or install on the HDD, when I do either of these, there is a bunch of text on a black background, as I would expect, but after a few seconds the screen goes blank and the monitor says No signal.  I have previously installed Ubuntu with a flash drive to fix issues caused my Windows and ran into no problems.  Is it a problem with the pen drive I am using? Or maybe a problem with the system? I tried it in my USB 3.0 ports at the back and the USB 2.0 ones as well.  My Specs are:   &nbsp;&nbsp;AMD Phenom II x4 965BE &nbsp;&nbsp;ASUS AMD Radeon HD 6670 &nbsp;&nbsp;ASUS M5-L75 USB3 (typing this from memory, it's an AM3+ ASUS board with USB3.0 in the AMD700 chipet or something) &nbsp;&nbsp;Corsair XMS3 DDR3 4GB(2x2GB) 1333MHz RAM    I ran the memtest program on the flash drive as well, and it came out clean. Should I use single or double colon notation for pseudo-elements? Receiving Stack Overflow's Chat Notification This is regarding the feature where we can chat regarding a question on SO. There is an automated process for creating the room from comments. It's a very useful feature in case the OP is not able to understand an answer and needs more clarification.  What I don't see is a notification system for chats where there has been an extended period of time between communication.   Let's say someone replied to my chat after a few days, in that case I do not receive any notification.   This happened to me only a few hours ago, where in I had replied to one question but the OP was not available at that point and he replied to me after some days. As I didn't check that chat I had no way of noticing if someone replied to my message and the 48h window after my last activity in that chat, during which I would have received the @ convention ping notification, had already expired.  We receive notification on posts, upvotes that we receive, so why not the chat rooms?   Should there be a notification center like the one for posts (or may be include this in the same center as posts)? Is there any utility to check whether my laptop battery is in healthy condition or not? Is there any utility available on Ubuntu to check whether laptop battery is in healthy condition or not? How can I make a part of a texture glow? Node Package Manager got corrupted in some way, now it cannot be installed I ran into a problem today when I decided to use npm update -g in a nodejs app directory. This was a bad decision, because it caused me to run into a .   The solution is to reinstall nodejs and npm. Unfortunately when I go to install npm using sudo apt-get install npm it doesn't work.  The full error in the terminal is here:  Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that  some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation: The following packages have unmet dependencies:  npm : Depends: nodejs but it is not going to be installed    Depends: node-abbrev (&gt;= 1.0.4) but it is not going to be installed    Depends: node-ansi but it is not going to be installed    Depends: node-archy but it is not going to be installed    Depends: node-block-stream but it is not going to be installed    Depends: node-fstream (&gt;= 0.1.22) but it is not going to be installed    Depends: node-fstream-ignore but it is not going to be installed    Depends: node-github-url-from-git but it is not going to be installed    Depends: node-glob (&gt;= 3.1.21) but it is not going to be installed    Depends: node-graceful-fs (&gt;= 2.0.0) but it is not going to be installed    Depends: node-inherits but it is not going to be installed    Depends: node-ini (&gt;= 1.1.0) but it is not going to be installed    Depends: node-lockfile but it is not going to be installed    Depends: node-lru-cache (&gt;= 2.3.0) but it is not going to be installed    Depends: node-minimatch (&gt;= 0.2.11) but it is not going to be installed    Depends: node-mkdirp (&gt;= 0.3.3) but it is not going to be installed    Depends: node-gyp (&gt;= 0.10.9) but it is not going to be installed    Depends: node-nopt (&gt;= 2.1.1) but it is not going to be installed    Depends: node-npmlog but it is not going to be installed    Depends: node-once but it is not going to be installed    Depends: node-osenv but it is not going to be installed    Depends: node-read but it is not going to be installed    Depends: node-read-package-json (&gt;= 1.1.0) but it is not going to be installed    Depends: node-request (&gt;= 2.25.0) but it is not going to be installed    Depends: node-retry but it is not going to be installed    Depends: node-rimraf (&gt;= 2.2.2) but it is not going to be installed    Depends: node-semver (&gt;= 2.1.0) but it is not going to be installed    Depends: node-sha but it is not going to be installed    Depends: node-slide but it is not going to be installed    Depends: node-tar (&gt;= 0.1.18) but it is not going to be installed    Depends: node-which but it is not going to be installed E: Unable to correct problems, you have held broken packages.   What can I do to get npm properly installed and running again?  Thanks. In the definition of , I read the following text:     Many of the act's provisions were to sunset beginning December 31, 2005, approximately 4 years after its passage. In the months preceding the sunset date, supporters of the act pushed to make its sunsetting provisions permanent, while critics sought to revise various sections to enhance civil liberty protections. In July 2005, the U.S. Senate passed a reauthorization bill with substantial changes to several sections of the act, while the House reauthorization bill kept most of the act's original language. The two bills were then reconciled in a conference committee that was criticized by Senators from both the Republican and Democratic parties for ignoring civil liberty concerns.   What does sunset mean? In which cases can the word be used with a similar meaning? How to create local total counters? I would like to write page numbers like this '7/9' in all minted environments of my document. The first number ('7') is the current page and the second number ('9') is the maximum number of page breaks the source file has been split into.  I was planning to use the package totcount but I realized that it was only considering the last value of the counter at the end of the document and it couldn't manage counters locally to an environment.  Here is a totally stupid example, but it illustrate the problem:  \documentclass{report}  \usepackage{totcount} \newtotcounter{sectnum}  \let\oldsection\section \renewcommand{\section}{%   \setcounter{sectnum}{0}%   \regtotcounter{sectnum}%   \oldsection% }  \let\oldsubsection\subsection \renewcommand{\subsection}{%   \stepcounter{sectnum}%   \oldsubsection% }  \begin{document}  \section*{One}  \subsection*{SectionOne} \thesectnum / \total{sectnum}  \subsection*{SectionTwo} \thesectnum / \total{sectnum}  \subsection*{SectionThree} \thesectnum / \total{sectnum}  \subsection*{SectionFour} \thesectnum / \total{sectnum}   \section*{Two}  \subsection*{SectionOne} \thesectnum / \total{sectnum}  \subsection*{SectionTwo} \thesectnum / \total{sectnum}  \end{document}   Gives:    Where we would have liked to have: '1/4', '2/4', '3/4', '4/4' and '1/2', '2/2'.  So, is there a way to get this kind of feature ? Display lost/negative reputation in topbar achievements When you get reputation, you get a +X green spot on your reputation feed telling you that you gained reputation and if you drop down the menu, you get more detail.  How about we make it that when you lose more than you got since the last time you checked, you see it red with a little -X.  For example, let's say you've just looked at the reputation bar so any old rep notification is gone and you ask a new question. 3 users downvote your question and 1 upvotes it. You will see it green because you gained 5 rep but you should see it red because you are at -1 rep.   This would help people improve their questions / answer by seeing, as soon as they get on the site, that they got negative feedback and that something needs to be checked / changed.    It shows here so it could very well be applied to the notification count: Closed form for $\sum_{k=0}^{m} {\binom {m}{k}} a^{k} (b+ck)^N$ Is there a closed form for the following?   $$\sum_{k=0}^{m}  {\binom {m}{k}} a^{k} (b+ck)^N$$      how about a pretty limit for large $b$.   I have tried using the binomial expansion for the $(b+ck)^N$ to convert the above expression to double sum and then tried to simplify the double sum which was not successful. Any idea if I have find a closed form for it? In a world without air, I understand they definitely would. However, with drag taken into account, I think they wouldn't. Since the drag force varies proportional to the square of speed (ignoring the change in coefficient of drag with speed), wouldn't the total impulse due to drag on a bullet fired from a gun from a specific height horizontal to ground be higher than the total impulse on an identical bullet dropped from the same height with no horizontal velocity?  Also, when bullet spin is taken into account, the bullet fired from a gun should resist change in orientation; therefore, it should maintain a small but non-zero angle of attack, and also have a lift force, correct? I have a Canon 7D, which has a crop factor of 1.6x.  When using a Canon EF-S lens I get the focal length stated on the lens, when using a Canon EF lens I get the focal length times 1.6 when used on the APS-C body. At least that's what I thought.  I just compared two lenses:   Canon EF-S 15-85mm f3.5-5.6 Canon EF 70-200mm f4   .  The top photo is shot with the 15-85, the bottom photo with the 70-200 - both lenses are set to 70mm. Now why do they look the same?  I thought I would get 112-320mm (70mm*1.6 crop factor). What did I misunderstand about the crop factor calculation? Lightroom also displays the photos as having "70mm" and not 112mm. SEO in Stack Overflow We've been out of private beta for only two days, and stackoverflow.com is already ranking high in .  Are there any specific techniques applied in the site? I've been researching the subject for some time and haven't found it to be so easy.   even returns site-links.  ()