We are given a whole number $n \geq 1$ and independent Random variables $X_n$ with distribution $Poi(n)$ and $Y$ with distribution $Poi(2n)$. Determine $\lim_{n \to \infty} P(X_n+n = Y_n)$.  I tried to write the probability as$\sum_{k=0}^{\infty}P(X_n=k)P(Y_n=n+k)$ but I didn't make progress. I suspect I need to use one of the limits laws (Central limit, large numbers, small numbers) but I am not sure how. Suggestions?
A pitcher contains $X + 1$ blue balls and $Y + 1$ red balls. It is known that $X, Y$ are independent random variables, and it is given that $X \sim \mathrm{Poisson}(n), Y \sim\mathrm{Poisson}(2n)$. Calculate the limit: $\displaystyle\lim_{n\to \infty}\mathbb P (X + n = Y) $   I tried to condition on the value of $Y$ and to use the Law of total probability but it did not lead to a solution. ( I tried to sum: $ \sum_{i=0}^\infty\mathbb P(X+n=Y|Y=i)\mathbb P(Y=i) $ and get an expression that I can calculate its limit as n goes to infinity)
I would like to install the restricted extras package, which I didn't select during the install. How can I do that?        is a meta-package that installs:         Support for MP3 and unencrypted DVD playback   Microsoft TrueType core fonts   Flash plugin   codecs for common audio and video files Normally, it takes three votes to delete.  It still does, though the mouseover tip for a question with one delete vote cast on it already appears to be stuck at '1'    And then after casting a delete vote on it:    And if I've cast the first delete vote:    The above examples are from P.SE, and while I don't have the privs to verify this being the situation on other sites, this has been confirmed on at least SO and is likely network wide.    Child meta dups:   SO: I want to prove the assertion:      The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 \pmod{4}$.   My first attempt is this. In $\mathbb{Z}[\zeta_p]$, $1-\zeta_p$ is prime and $$ p = \epsilon^{-1} (1-\zeta_p)^{p-1} $$ where $\epsilon$ is a unit.  Since $p$ is an odd prime, $(p-1)/2$ is an integer and  $$ \sqrt{\epsilon p } = (1-\zeta_p)^{(p-1)/2} $$ makes sense and belongs to $\mathbb{Z}[\zeta_p]$.    How do I deal with the $\epsilon$ under the square root?  I guess the condition on the congruence class of $p$ comes from that.  Is this even the right way to proceed?  Uniqueness is not clear to me either.  I thought about looking at the valuation $v_p$ on $\mathbb{Q}$, extending it to two possible quadratic extensions beneath $\mathbb{Q}(\zeta_p)$, then seeing how those have to extend to common valuations on $\mathbb{Q}(\zeta_p)$, but I didn't see how to make it work.  I would appreciate some help. Does the mass of a battery change when charged/discharged? If so, how much? Is it possible to detect it, or is it beyond any measurement?  I'd say there are two possible scenarios (depending on the battery type) and both seem interesting:   The battery reacts chemically with its environment. The battery doesn't exchange any matter with its environment except electrons. I suppose there should be some difference at least due to the principle of energy-matter equivalence, but the difference is most likely immeasurable. Apt sources.list problem , packages not found? I am trying to install WICD but getting following errors on my ubuntu 10.04 box  sudo apt-get install wicd   Err http://archive.ubuntu.schoolnet.lk/ubuntu/ lucid/universe python-wicd 1.7.0+ds1-2   Could not connect to archive.ubuntu.schoolnet.lk:80 (103.29.60.18). - connect (111: Connection refused) Err http://archive.ubuntu.schoolnet.lk/ubuntu/ lucid/main python-iniparse 0.3.1-1   Unable to connect to archive.ubuntu.schoolnet.lk:http: Err http://archive.ubuntu.schoolnet.lk/ubuntu/ lucid/universe wicd-daemon 1.7.0+ds1-2   Unable to connect to archive.ubuntu.schoolnet.lk:http: Err http://archive.ubuntu.schoolnet.lk/ubuntu/ lucid/universe wicd-gtk 1.7.0+ds1-2   Unable to connect to archive.ubuntu.schoolnet.lk:http: Err http://archive.ubuntu.schoolnet.lk/ubuntu/ lucid/universe wicd 1.7.0+ds1-2   Unable to connect to archive.ubuntu.schoolnet.lk:http: Failed to fetch http://archive.ubuntu.schoolnet.lk/ubuntu/pool/universe/w/wicd/python-wicd_1.7.0+ds1-2_all.deb  Could not connect to archive.ubuntu.schoolnet.lk:80 (103.29.60.18). - connect (111: Connection refused) Failed to fetch http://archive.ubuntu.schoolnet.lk/ubuntu/pool/main/p/python-iniparse/python-iniparse_0.3.1-1_all.deb  Unable to connect to archive.ubuntu.schoolnet.lk:http: Failed to fetch http://archive.ubuntu.schoolnet.lk/ubuntu/pool/universe/w/wicd/wicd-daemon_1.7.0+ds1-2_all.deb  Unable to connect to archive.ubuntu.schoolnet.lk:http: Failed to fetch http://archive.ubuntu.schoolnet.lk/ubuntu/pool/universe/w/wicd/wicd-gtk_1.7.0+ds1-2_all.deb  Unable to connect to archive.ubuntu.schoolnet.lk:http: Failed to fetch http://archive.ubuntu.schoolnet.lk/ubuntu/pool/universe/w/wicd/wicd_1.7.0+ds1-2_all.deb  Unable to connect to archive.ubuntu.schoolnet.lk:http: E: Unable to fetch some archives, maybe run apt-get update or try with --fix-missing? Can I make my Ubuntu make sounds when I press Caps Lock etc.? I am a UI designer and developer and have been designing websites for aound 10 years. I want to get into user experience design. I know how to do a design, but I am lagging the terminologies and the technical things in the user experience field.   So how do I go about getting the skills and mindset necessary for being a User Experience Designer? Why aren't orbitals symmetric? Need help with a question regarding the Big Oh In class we are currently covering upper/lower bounds, big Oh and omega and the like. I am pretty good on the "typical" functions one would do, but at a complete loss at "general" statements. This ons function, for example:  $$\forall f\in\mathcal{F}: \lfloor \sqrt{\lfloor f(n)\rfloor }\rfloor \in O(\sqrt{f(n)}).$$  the domain F by the same is all natural numbers that return a real positive number when inputted. My first instinct with something like this is to remove the floors, the general strategy I though to solve this is remove the outer floor, then get rid of the square root, and then the inner floor. But I have no idea how to go about this or if this is even the right way to do it.  Help please? Thanks in advance. SEO for websites with very less content and a lot of images Let's say we have an image exhibition website that has got a lot of images but next to nothing content. The focus keyword for this website is "Artistic wallpapers". What would be the best approach w.r.t SEO for such a website.  Here are a few things I can imagine:   Submitting sitemaps ( image and web ) Having a good pagespeed score. Nice navigation Breadcrumbs sharing the website content on Social media. Having appropriate titles.   Having worked on all these things, what can be done if Search engines are picking wrong keywords ( For example views, votes, ratings) Can anything be done so that Search engines don't track these specific items? Also, is it a good practice to have all the pages submitted to Search engines at once? How to stop iPhone users from sending iMessages and send SMS instead I used to have an iPhone, but I changed to an android (shock, horror!). Unfortunately Apple refuses to let me go and whenever friends message me the messages get sent as iMessages (unless they remember to specifically chnge the format). I never get these messages. They never get resent as SMSs automatically as is supposed to happen. How can I get my phone number deleted from the iMessage list. I've turned off iMessage on the old iPhone. C# 3.0 auto-properties — useful or not? Note: This was posted when I was starting out C#. With 2014 knowledge, I can truly say that auto-properties are among the best things that ever happened to the C# language.  I am used to create my properties in C# using a private and a public field:  private string title; public string Title {     get { return title;  }     set { title = value;  } }   Now, with  3.0, we got auto-properties:  public string Title { get; set; }   I know this is more a philosophical/subjective questions, but is there any reason to use these auto-properties except from saving five lines of code for each field? My personal gripe is that those properties are hiding stuff from me, and I am not a big fan of black magic.  In fact, the hidden private field does not even show up in the debugger, which is OK given the fact that the get/set functions do nothing. But when I want to actually implement some getter/setter logic, I have to use the private/public pair anyway.  I see the benefit that I save a lot of code (one vs six lines) without losing the ability to change the getter/setter logic later, but then again I can already do that by simply declaring a public field "Public string Title" without the need of the { get; set; } block, thus even saving more code.  So, what am I missing here? Why would anyone actually want to use auto-properties? I want to mount the device /dev/sda3 to the directory /foo/bar/baz. After mounting the directory should have the uid of user johndoe. So I did:  sudo -u johndoe mkdir /foo/bar/baz stat -c %U /foo/bar/baz johndoe   and added the following line to my /etc/fstab:  /dev/sda3 /foo/bar/baz ext4 noexec,noatime,auto,owner,nodev,nosuid,user 0 1   When I do now sudo -u johndoe mount /dev/sda3 the command stat -c %U /foo/bar/baz results in root rather than johndoe. What is the best way to mount this ext4-filesystem with uid johndoe set? After doing a "sudo su -" on an Ubuntu 12.04 notebook I did a "crontab -e", added this:   * * * * * env DISPLAY=:0.0 /usr/bin/gnome-calculator   and waited for minutes. Nothing Happened. I don't have any external monitors and if I run this command "env DISPLAY=:0.0 /usr/bin/gnome-calculator" in the terminal, it just works. But not from cron. Why?   The syslog only contains this:   May 24 14:37:01 localhost cron[1227]: (root) RELOAD (crontabs/root) May 24 14:37:01 localhost CRON[16432]: (root) CMD (env DISPLAY=:0.0 /usr/bin/gnome-calculator )   And I already tried an "xhost +localhost".   [root@NOTEBOOK /var/log] xhost access control enabled, only authorized clients can connect INET:localhost.localdomain SI:localuser:USERNAME [root@NOTEBOOK /var/log]    So the solution for another question like this on askubunut didn't worked. What does the |= operator do in Java? Some images shown outside of preview rectange Some images shown outside of preview rectange before uploading, eg: Preventing robots from crawling specific part of a page As a webmaster in charge of a tiny site that has a forum, I regularly receive complains from users that both the internal search engine and that external searches (like when using Google) are totally polluted by my users' signatures (they're using long signatures and that's part of the forum's experience because signatures makes a lot of sense in my forum).  So basically I'm seeing two options as of now:   Rendering the signature as a picture and when a user click on the "signature picture" it gets taken to a page that contains the real signature (with the links in the signature etc.) and that page is set as being non-crawlable by search engine spiders).  This would consume some bandwidth and need some work (because I'd need an HTML renderer producing the picture etc.) but obviously it would solve the issue (there are tiny gotchas in that the signature wouldn't respect the font/color scheme of the users but my users are very creative with their signatures anyway, using custom fonts/colors/size etc. so it's not that much of an issue). Marking every part of the webpage that contains a signature as being non-crawlable.   However I'm not sure about the later: is this something that can be done?  Can you just mark specific parts of a webpage as being non-crawlable? How do I install Ubuntu? How does the eye perform the equivalent of white balance? Possible Duplicate:          How does the eye perform the equivalent of white balance? Why is the camera not able to replicate it? Norm of adjoint operator in Hilbert space