I have come across two proofs for this on , but I cannot wrap my head around the intuition behind this. Why should this hold?  I am looking for a possible geometric interpretation of the area under the curve.
I can see by manipulating the expression why $\mathbb{E}X$ works out to be $\int_0^\infty 1-F(x)\,dx$, where $F$ is the distribution function of $X$, but what is an intuitive explanation for why that is true? If at each point we sum the probability $\mathbb{P}(X&gt;x)$, why should we end up with the expectation?  Thanks
Generating all Pythagorean triplets for given Hypotenuse can anybody help me out to find all  Pythagorean triplet when Hypotenuse is given? for example 10 is given i need (10,6,8)  ,and not needed such triplet (10,24,26) as 10 is not Hypotenuse for this triplet! How will people upgrade from 12.10 to later releases after 13.04 is EOL? Detect edits to add [SOLVED] or [RESOLVED] to the title and direct the user to add an answer Users often ask questions that they answer for themselves. Good for them. And good for us.  Symptom  The correct procedure seems to be:   Write new answer to own question  Wait two days Accept own answer   The incorrect, but fairly popular, procedure seems to be:   Edit the question to include the solution Put "[SOLVED] or [RESOLVED]" in the title   Proposed solution  Users should be warned when using the word "solved", delimited in any way, in a question title. This could be "[SOLVED]", "(solved)", or even "Solved - ". I imagine this:   The user tries to submit an edit that adds "solved" such that the word "solved is delimited. The user is brought to a warning page. If the only edit to the question was appending new lines (such as adding an "edit:" paragraph), that edit is copied into a text box The page mentioned in 1 is shown as follows:     $StackSite marks questions as answered when you accept an answer. You can accept an answer by clicking the green check mark next to that answer. There's no need to put "SOLVED" or "RESOLVED" in the title!      Are you answering your own question?      Create a proper answer to your question:         Move your edit to a new answer.   Write your answer.   If nobody writes a more correct or thorough answer after two days, come back and accept your answer.         By doing this, you remove your question from the Unanswered Questions list. You also allow other $StackSite users to vote your answer up. Thanks for helping out!      TEXT BOX WITH EDITS GOES HERE, FOLLOWED BY "POST YOUR ANSWER" BUTTON      Not answering your own question?      Sorry for slowing you down.      OVERVIEW OF EDIT, FOLLOWED BY "SAVE EDITS" BUTTON    Discussion  This would likely reduce the number of [RE]SOLVED questions that appear. However, it's unclear whether this should be a community moderation issue or a feature request issue. The increased engineering work required in building in the interstitial page is probably quite nontrivial, whereas this type of question makes up well under .  Note: Google also provides a . Proving $1-\exp(-4x^2/ \pi) \ge \text{erf}(x)^2$ This is a follow-up to  question. I came across a tighter potential bound and checked it numerically for $0\le x\le 5$. I think it holds for all positive $x$, can anyone see a proof?  $$1-\exp(-4x^2/ \pi) \ge \text{erf}(x)^2$$  Note: using analysis for previous  you can show that $1-\exp(-k x^2)$ is an upper bound on $\text{erf}(x)^2$ for $k=2$ and a lower bound when $k=1$. The factor $k=4/\pi$ comes out when numerically searching for tightest upper bound. Not only does it seem to give an upper bound, but it also tracks $\text{erf}(x)^2$ very closely.  Dashed graph below is $\text{erf}(x)^2$, red is $1-\exp(-k x^2)$ for $k=4/\pi$, other two graphs are for $k=1$ and $k=2$ If a student needs a reference letter for graduate admissions or that sort of thing, some profs will ask the student to write a reference letter for themself before sending it off to the prof for minor edits and finalization. This seems to be a somewhat common practice given that some graduate schools ask for several letters of recommendation even when it is not reasonable for the student to have developed deep connections with that many professors, and that most professors are just too busy to write quality letters for all the students that ask them to.  Is it acceptable to write most of the reference letter and have the prof make minor edits? Do academic institutions frown upon this practice? Would it be considered an academic offense if a student wrote a reference letter for themselves and had a prof sign it? Can I use an nVidia Geforce 9600GT for triple-monitor? Possible Duplicate:          My work PC has (as you can guess from the title) a single Geforce 9600GT card. According to the  for this card, both DVI ports are Dual-Link DVI. Both are currently in use on Dell 19" 1280x1024 screens. I'm running Windows 7, with Aero enabled, for what that's worth.  What I'm wondering is if I can use a standard DVI splitter cable to drive a third screen from one of the ports -- specifically a Samsung 22" 1680x1050. I'm really outgrowing the two-screen setup, and am strongly considering bringing in my home monitor (which is almost never used) to bolster my usable space. If I can't, then my alternative is to replace one of these Dells with that Samsung, to give me at least a "little" more space.  (Before you ask, there's probably no chance I can get my company to spring for a second video card and third monitor. Even if I could, it would not be politically expedient, as a number of my other coworkers are already pretty jealous at my setup.) In "Seeing like a state" of James C.Scott there is a sentence at the beginning of a paragraph (Acknowledgements xi):     There are a good many scholars whose writings opened up new   perspectives for me or provided outstanding analyses of issues that I   could not have hoped to study so comprehensively on my own.   What worries me is a good many scholars. Is this some type of inversion or ancient English? Is this grammatically right? Why is a here? I thought that appropriate version would be There are many good scholars. Am I wrong? Array index out of bound behavior Why does C/C++ differentiates in case of array index out of bound #include &lt;stdio.h&gt; int main() {     int a[10];     a[3]=4;     a[11]=3;//does not give segmentation fault     a[25]=4;//does not give segmentation fault     a[20000]=3; //gives segmentation fault     return 0; }  I understand that it's trying to access memory allocated to process or thread in case of a[11] or a[25] and it's going out of stack bounds in case of a[20000]. Why doesn't compiler or linker give an error, aren't they aware of the array size? If not then how does sizeof(a) work correctly? What does "...from nonmagical attacks not made with silvered weapons" mean? I'm sure your first thought is, that's perfectly clear.  Unfortunately my group who aren't normally rules lawyers disagree.  Party encounters a  (MM p.211).  It has:     Damage Immunities: bludgeoning, piercing, and slashing from nonmagical   attacks not made with silvered weapons.   My reading of that is if the weapon has to be magical and silvered to hurt it.  So a +1 silver dagger will hurt it, but a +1 regular dagger or a +0 silver dagger won't.  The group's interpretation was that all three of those weapons would hurt and only a +0 regular dagger would be ineffective.    I would also note my interpretation would mean something like blade barrier would also not harm it since you could argue they're magical blades but nothing indicates that they're silvered.    This also appears to be the same wording for some undead so I'm not looking for a lycanthrope specific interpretation but something general for every time that wording is used.    Thanks in advance. We are using .Net to make web service calls the Salesforce API. Is there a way to know for sure that the calls are using TLS 1.2? Are there any API logs or records of the protocol is use in Salesforce we can review to be confident the change is working? Can I deduct a contribution to a traditional IRA if I am also contributing to a 401(k) in the same tax year? How to obtain the sum of the following series? $\sum_{n=1}^\infty{\frac{n^2}{2^n}}$ I am looking for a way to replace a string in a file with a string that contains a slash by using sed.  connect="192.168.100.61/foo" srcText="foo.bar=XPLACEHOLDERX" echo $srcText | sed "s/XPLACEHOLDERX/$connect"   The result is:  sed: -e Expression #1, Character 32: Unknown option for `s' What are attention mechanisms exactly? Attention mechanisms have been used in various Deep Learning papers in the last few years. Ilya Sutskever, head of research at Open AI, has enthusiastically praised them:    Eugenio Culurciello at Purdue University has claimed that RNNs and LSTMs should be abandoned in favor of purely attention-based neural networks:    This seems an exaggeration, but it's undeniable that purely attention-based models have done quite well in sequence modeling tasks: we all know about the aptly named paper from Google,   However, what exactly are attention-based models? I've yet to find a clear explanation of such models. Suppose I want to forecast the new values of a multivariate time series, given its historical values. It's quite clear how to do that with an RNN having LSTM cells. How would I do the same with an attention-based model? What happened to Startups SE? I really enjoyed that site. I've only posted on it once, but I enjoyed seeing other people's posts and appreciated reading about entrepreneurial dilemmas. Why did the quality of terminators increase from movies 1 to 3? I have seen only the first three movies, so I don't know whether the other movies/TV series deal with this question or not. My main point is that in the movies, the terminator that travels back furthest in time is of the worst quality, and the quality increases with the time travel (meaning they travelled from a later time). What is a good explanation for this?  Note: I am not interested in the obvious not-in-universe explanation that they wanted something better and bigger for each new movie. How to force a web browser NOT to cache images Background  I am writing and using a very simple CGI-based (Perl) content management tool for two pro-bono websites. It provides the website administrator with HTML forms for events where they fill the fields (date, place, title, description, links, etc.) and save it. On that form I allow the administrator to upload an image related to the event. On the HTML page displaying the form, I am also showing a preview of the picture uploaded (HTML img tag).  The Problem  The problem happens when the administrator wants to change the picture. He would just have to hit the "browse" button, pick a new picture and press ok. And this works fine.  Once the image is uploaded, my back-end CGI handles the upload and reloads the form properly.  The problem is that the image shown does not get refreshed. The old image is still shown, even though the database holds the right image. I have narrowed it down to the fact that the IMAGE IS CACHED in the web browser. If the administrator hits the RELOAD button in Firefox/Explorer/Safari, everything gets refreshed fine and the new image just appears.  My Solution - Not Working  I am trying to control the cache by writing a HTTP Expires instruction with a date very far in the past.  Expires: Mon, 15 Sep 2003 1:00:00 GMT   Remember that I am on the administrative side and I don't really care if the pages takes a little longer to load because they are always expired.  But, this does not work either.  Notes  When uploading an image, its filename is not kept in the database. It is renamed as Image.jpg (to simply things out when using it). When replacing the existing image with a new one, the name doesn't change either. Just the content of the image file changes.  The webserver is provided by the hosting service/ISP. It uses Apache.  Question  Is there a way to force the web browser to NOT cache things from this page, not even images?  I am juggling with the option to actually "save the filename" with the database. This way, if the image is changed, the src of the IMG tag will also change. However, this requires a lot of changes throughout the site and I rather not do it if I have a better solution. Also, this will still not work if the new image uploaded has the same name (say the image is photoshopped a bit and re-uploaded). Objective C const NSString * vs NSString * const Seating of $n$ people with tickets into $n+k$ chairs with 1st person taking a random seat "$n$ music lovers have reserved seats in a theater containing a total of $n+k$ seats ($k$ seats are unassigned). The first person who enters the theater, however, lost his seat assignment and chooses a seat at random. Subsequently, people enter the theater one at a time and sit in their assigned seat unless it is already occupied. If it is, they choose a seat at random from the remaining empty seats. What is the probability that person $n$, the last person to enter the theater, finds their seat already occupied?"  I could do this problem for small specific values of $n$ and $k$ but as they grow the expressions seem to get really messy really quick with no discernible pattern. How would one solve this problem? Best compiler warning level for C/C++ compilers?