Let $X$ and $Y$ be two random variables with finite second moments and $E(X \mid Y) = Y$ , $E(Y \mid X)= X$. Show that $P(X = Y)= 1$. I tried using conditional expectation, so $E(E(X \mid Y))=E(Y) \Rightarrow E(X)=E(Y)$, but then I'm totally stuck on how to continue. I don't even know how to relate it to probability or using the fact of finite second moment. Any help is appreciated. Thank you.
Assume that $X$ and $Y$ are two random variables such that $Y=E[X|Y]$ almost surely and $X= E[Y|X]$ almost surely. Prove that $X=Y$ almost surely.   The hint I was given is to evaluate: $$E[X-Y;X&gt;a,Y\leq a] + E[X-Y;X\leq a,Y\leq a]$$  which I can write as: $$\int_A(X-Y)dP +\int_B(X-Y)dP$$ where $A=\{X&gt;a, Y\leq a\}$ and $B=\{X\leq a,Y\leq a\}$.  But I need some more hints.
Buy a second seat in order to bring an extra bag? What will Sony do if someone uses my account as primary on another ps4 without my knowledge? What will Sony do if someone uses my account as primary on another ps4 without my knowledge? Will the account be banned? Or will the other guy that has my account as primary be banned? After how long? I have no credit card on my account so I don't care too much but I want to know.   He uses my account on his ps4 as a primary account. I think to use my digital games as free (Sharing, but without my knowledge). This is the announcement post of the Stack Exchange Data Explorer. Its answers were meant to iron out the initial glitches 6 to 8 years ago.  If you have found a bug or need support  today with any of SEDEs features, please ask a new question and tag it with   What does it do? Allows you to run arbitrary SQL queries on the data dumps, and share those queries with your friends. Technical details It is hosted on the Stack Exchange infrastructure , the database is hosted on the same SQL Server version all other sites are hosted on. You can use most of the TSQL commands, even create temporary objects (but not User Defined Functions). The controls used are jQuery based, the grid is  . All query results are cached, so sharing a link to a query will not result in the server going in to a tailspin. The query edit control is a slightly modified  control, I also am considering bespin cause the inter iframe communication is nasty complicated. It gives up nice syntax highlighting for queries. Parameters There is support for parameterized queries, to get it to work name your params like ##this##. For example have a . Query name and description A query can have a name and description, to name it lead with a comment, to describe it continue commenting, for example: -- This is my query name  -- This is my description -- I can span multiple lines  SELECT 1  Anonymous users can initially name a query, but can not change the names of any existing queries. Tutorial There's a  which helps you getting started in writing your own SQL queries. Logging in You can use your openid provider to log in, at the moment once logged in we will track the list of queries you execute (giving you basic history support) Also if there are any params named UserId they will be populated automatically if your EmailHash matches the one on the site being queried. Magic columns If you alias an id column with as [Post Link] it will automatically create a link in the result set to the parent site. Similarly if you alias an id column with as [User Link] it will display a link to the user page. select top 10 * from Posts will show you how tags are done. (Magic columns for images is planned) example (most controversial posts:  ) So what are you waiting for, check it out at  Any feedback would be greatly appreciated. I hope to have the login section working fairly soon and allow voting on the best queries and commenting. Featured Queries At the moment, admins have permission to feature interesting queries on this page , I am looking for better ways to manage the huge query list that is building up. Can HTML5 databases and localStorage be shared across subdomains? I was having a time getting my bootable USB 3, 32GB W10 install USB to be recognized. It recognized one time but I had to restart for something and it never came up again. I eventually removed all other boot up options to make it easier and faster to try again (no F12 needed).  Finally I gave up and tried another drive (16GB this time FWIW) and then it worked.  HOWEVER, it's stuck on a reboot loop now. I can choose to install, then to select the right disk, then it says it's loading file and then next thing I know it's saying do you want to install or upgrade. Covering map between orientable compact connected surfaces without boundary with genus $g$ Similar to a previous question I asked, this is the other question that I couldn't answer in an exam I gave for my Topology course:  Suppose $g_1,g_2$ are non-negative integers such that $2-2g_2$ divideds $2-2g_1$. I have to construct a covering map $S_{g_1} \rightarrow S_{g_2}$, where $S_g$ is denotes an orientable compact connected surface without boundary with genus $g$.  Now, I attempted to try this problem for surfaces with smaller genera, for example a 2-torus and a 2-sphere. Clearly $2-2g$ is the Euler Characteristic of the said surface, and by classification of surfaces we have to only consider connected sum of 2-tori. So, I have $\chi(S^2)=2$, and $\chi(T^2) = 0$, here we have $2$ dividing $0$, hence I have to construct a covering map from $T^2 \rightarrow S^2$, and I have no idea how to move forward.  Addendum: I have to got great answers to my question here but I had a doubt when it comes to the points being mapped, specifically in Giulio Bresciani's answer. Clearly the handles of the figure is mapped to the handles in the figure above, but I'm unable to understand the mapping of the twist of the cover (above figure) to the extreme handles of the base (the figure below). Any help? I am struggling to find such languages $L_{1}$, $L_{2}$, and $L_{3}$ such that  $$ L_{1} \subseteq L_{2} \subseteq L_{3} $$ where $L_{1}, L_{3} \notin  \mathbb{R}$ and $L_{2} \in  \mathbb{R}$.  I know they exist, I need help finding them. How to set Google+ photos to be Public-Unlisted (as you could do with PicasaWeb)? Previously, if I uploaded photos from Picasa to PicasaWeb, I could specify them as Public but Unlisted. So you had to have the link to view them but no login was required.  I don't see that option now that Picasa uploads to Google+.  Any ideas? Let $\mathbf{Y}$ be a random vector. Are $k$th moments of $\mathbf{Y}$ considered? I am self-learning on linear model theory right now, and one thing I find surprising is that although $\mathbb{E}[\mathbf{Y}]$ is defined for a random vector $\mathbf{Y} = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n\end{bmatrix}$, there is no mention of further moments besides the covariance matrix.  Google searching hasn't turned up much. Are $k$th (raw) moments of $\mathbf{Y}$ considered, or is there a different idea I don't know about?  I am learning from the text  (the TOC starts in p. 17 of the linked file). By "considered," what I mean is is there such a thing as $\mathbb{E}\left[\mathbf{Y}^k\right]$, and if so, how would such a concept be defined? The book I have only covers the first raw moment, and I find it a bit strange that there is no mention of how to define $\mathbb{E}\left[\mathbf{Y}^k\right]$ given my experience in univariate probability, nor do I have the expertise to define it.  Furthermore, if $\mathbb{E}\left[\mathbf{Y}^k\right]$ isn't defined, is there perhaps a related concept that I don't know about that is used instead? What are the benefits of running chef-server instead of chef-solo? I am looking at automated deployment solutions for my team and have been playing with Chef for the past few days. I've been able to get a simple web app up an running from a base Red Hat VM using chef-solo.  Our end goal is to use Chef (or another system) to automatically deploy application topologies to the cloud as we run builds. Our process would basically run like this:   Our web app code, dependencies, and chef cookbooks are stored in SCM A build is executed and creates a single package for images to acquire and test against The build engine then deploys new cloud images that run a chef client to get packages installed. The images acquire the cookbooks from SCM or the Chef server and install everything to get up and running   What are the benefits and/or use cases for getting a Chef Server running?  Are there any major benefits to have a Chef Server hold and acquire the cookbooks from SCM vs. using chef-solo and having a script that will pull the cookbooks from SCM? Gmail App in my Android device shows correctly all the labels that I created using Gmail through web interface.  Can I create new labels in Gmail on Android device using Gmail App?   Also, ideally, once I create these new labels in Gmail App, I obviously should be seeing the same in Web Interface of Gmail too (Sync!!).  Any ideas? Does the lens I just bought have a fungus? This equation is directly given in my book and I am don't know anything about its proof.I tried L'Hospital rule by differentiating the both numerator as well as denominator(division rule), but the result is still coming in indeterminate forms.I am a beginner , and haven't practiced limits that much. This formula is really confusing me. Temperature and kinetic energy of molecules I was wondering if temperature is related to the average translational kinetic energy of the molecules then why does the average kinetic energy of say a moving object not affect the temperature? What is the mechanism regarding these vibrational motions that imparts temperature. Also do a gas and liquid at the same temperature possess the same kinetic energy? and a heavier molecule traveling at the same speed as a lighter molecule will have a greater temperature? I have three displays, a 4K display flanked by two 1080p displays. Physically, these monitors are roughly the same size. Furthermore, Windows 10 handles the DPI scaling of the 4K monitor separately from the 1080p monitors just fine. So while it appears as such to Windows    the real layout is three monitors of equal size side by side.  The problem is that the mouse keeps getting caught on the edges of display 2 and three when dragging from display 1. I would like the mouse to continue onto display 2 or 3 regardless of which part of the left or right edge of display 1 the mouse touches.  This question has been asked before (e.g. ) and a common answer is to suggest that this behaviour is intentional on the basis that mouse movement should reflect different size displays. Such an answer is not appropriate as it doesn't consider DPI and scaling. I recently renewed my passport. I also have an expired passport which has my Canada visa (visitor's) valid for another three years. I have already used my Canadian visa which is multiple entry. I am planning to visit Canada this summer.  Since my valid Canadian visa is in my old/expired passport, do I need to renew/transfer my Canada visa or can I simply travel with valid visa in old passport? Nvidia driver updated - mixed versions of 304.88 and 319.32 My problem is that after trying to update my driver through the additional drivers GUI I can't boot Ubuntu with GUI anymore, since Xorg says the kernel module is of version 304.88 (The old version before I updated) but the driver components are of version 319.32 (The new version)  This is the output of dpkg -l | grep nvidia  dkpg -l | grep nvidia ii nvidia-304 : 304.88-0ubuntu0.0.3  NVIDIA binary Xord driver, kernel module and VDPAU library ii nvidia-319 : 319.32-0ubuntu0.0.1  NVIDIA binary Xord driver, kernel module and VDPAU library ii nvidia-common : 1:0.2.44.2  Find obselete NVIDIA drivers ii nvidia-current : 304.88-0ubuntu0.0.3  Transitional package for nvidia-current ii nvidia-settings : 304.88-0ubuntu.0.0.3  Transitional package for nvidia-settings ii nvidia-settings-304 : 304.88-0ubuntu.0.0.3  Tool for configuring the NVIDIA graphics driver ii nvidia-settings-319 : 319.32-0ubuntu.0.0.1  Tool for configuring the NVIDIA graphics driver   That's the output of that file.  The message startx gives me is:  NVIDIA: API mismatch: the NVIDIA kernel module has version 304.88, but this NVIDIA driver component has version 319.32. Please make sure that the kernel module and all NVIDIA driver components have the same version.     Fatal server error:    no screens found   I understand that something went wrong during the update, so I wonder how to fix this so that the remaining parts of 304 are removed and replaced with 319. I don't know hwere to start because I thought that nvidia-current was the one that would update to 319 and so reinstalling that didn't work. And I googled and found that somebody said to try to install/reinstall/uninstall this package: nvidia-current-updates-dev  But I didn't have that package installed so it did not help. Other than that, I am clueless. I hope the information I provided will be enough so I can get an answer soon.  EDIT:  lsb_release -a No LSB modules are available. Distributor ID: Ubuntu Description: Ubuntu 12.04.3 LTS Release: 12.04 Codename: precise  uname -a Linux SQuad 3.2.0-52-generic #78-Ubuntu SMP Fri Jul 26 16:21:44 UTC 2013 x86_64 x86_64 x86_64 GNU/LINUX In JavaScript, the "this" operator can refer to different things under different scenarios.   Typically in a method within a JavaScript "object", it refers to the current object.  But when used as a callback, it becomes a reference to the calling object.  I have found that this causes problems in code, because if you use a method within a JavaScript "object" as a callback function you can't tell whether "this" refers to the current "object" or whether "this" refers to the calling object.  Can someone clarify usage and best practices regarding how to get around this problem?     function TestObject() {             TestObject.prototype.firstMethod = function(){                       this.callback();                       YAHOO.util.Connect.asyncRequest(method, uri, callBack);              }              TestObject.prototype.callBack = function(o){               // do something with "this"               //when method is called directly, "this" resolves to the current object               //when invoked by the asyncRequest callback, "this" is not the current object               //what design patterns can make this consistent?               this.secondMethod();             }             TestObject.prototype.secondMethod = function() {              alert('test');             }         } How to remove Ubuntu and put Windows back on? I have absolutely no experience with Linux, and I desperately need to get my computer back up and running again with Windows.  How do I remove Ubuntu and reinstall Windows?    Editor's note: many of the answers are about removing Ubuntu from dual-boot but keeping Windows (which is a bit complicated), while other answers are about removing Ubuntu from single-boot (which is easy: basically just format the disk while installing Windows). The question as written is ambiguous between dual-boot or single-boot. What is the word that is used to describe the oversimplification or non-academic discussion of a complex issue?   Typically, I'm looking for a word like pedantic, or puerile, except that the word covers the fact that the discussion is particularly non-academic. I know that such a word exists since I have used it before, but it has skipped my mind.