It is well known that if $X \sim B(m, p)$ and $Y \sim B(n, p)$ are independent then $X+Y \sim B(m+n, p)$ but what is the distribution of $X-Y$?  Here is what I have tried. $\Pr[X-Y = c] = \sum_{i=0}^n \Pr[X = c+i] \Pr[Y = i]$ which can be simplified to $\left(\frac{p}{1-p}\right)^c (1-p)^{m+n} \sum_{i=0}^n \binom{m}{c+i} \binom{n}{i} \left(\frac{p}{1-p}\right)^{2i}$. Letting $q = \left(\frac{p}{1-p}\right)^2$, simplifying this expression reduces to simplifying $\sum_{i=0}^n \binom{m}{c+i} \binom{n}{i} q^i$. But I am stuck here and perhaps there is a better approach.
Let $X$ ~ $Bin(n,p)$ and $Y$ ~ $Bin(m,p)$ be two independent random variables. Find the distribution of $Z=X-Y$.   see also   I figured this out:  $$ P(Z=z)=\cases{\sum_{i=o}^{min(m,n)} Bin(k+i,n,p)*Bin(i,m,p), &amp;if $z\ge0$;\cr \sum_{i=0}^{min(m,n)} Bin(i,n,p) * Bin(i-z, m, p),&amp;otherwise. \cr}$$  I also validated it by Monte Carlo simulation. For $n=30$, $m=20$ and $p=0.5$, I get the following distribution, where the circles are the analytical probabilities and the line connects the MC estimates.    Because that looked to me pretty much like a binomial distribution, I gave it a try and figured out that its actually a binomial, just shifted by m to the left. This can be simply written as $P(Y=y) = Bin(y+m, m+n, p)$. Hence, given equal success probabilities, the sum of two independent binomially distributed random variables is binomial, but also their difference, just shifted to the left.  This question here  is actually the same as mine, but received no answer and only the comment that there would be no simple formula. But the above formula looks pretty simple to me.   Is it correct that for the case of equal success probabilities, the above equations actually describes the distribution of $Z=X-Y$? I read in a book that $Z$ could not be binomial distributed because it had negative support. Is it right to call it a shifted binomial?
Forcing SSL and www in .htaccess I'm looking for a way to force SSL and www.  I've been able to force both separately but together I keep running into redirection issues. The following code works when handling a URL in this format: http://example.com and properly redirects to https://www.example.com but when the incoming URL is https://example.com it will not forward to https://www.example.com - Any suggestions?  EDIT: it should also send http://www.example.com to https://www.example.com.  RewriteCond %{REMOTE_ADDR} !127\.0\.0\.0 RewriteCond %{SERVER_PORT} 80 RewriteCond %{HTTP_HOST} !^www.example\.com$ RewriteRule ^(.*)$ https://www.example.com/$1 [R,L] Proof that every polynomial of odd degree has one real root I want to prove that every real polynomial of odd degree has at least one real root, using  the intermediate value theorem.  Let $P(x) = x^{2n+1} + a_n x^{2n} + . . . + a_0$ for each $a_i \in \mathbb{R}$ and $n \in \mathbb{N}$.  By the fundamental theorem of algebra I know that $P(x)$ has exactly $2n+1$ complex roots, so  $P(x) = (x+r_1)(x+r_2) . . . (x+r_{2n+1})$ for each $r_i \in \mathbb{C}$  I do not know how to complete this but I do know that, at some point, I probably have to show that each root with imaginary part non zero has to come in conjugate pairs, and since $2n+1$ is odd there is at least $1$ root that is imaginary part $0$ and thus real. Writing string at the same position using Console.Write in C# 2.0 I have a console application project in C# 2.0 that needs to write something to the screen in a while loop. I do not want the screen to scroll because using Console.Write or Console.Writeline method will keep displaying text on console screen incremently and thus it starts scrolling.  I want to have the string written at the same position. How can i do this?  Thanks Submitting a form by pressing enter without a submit button virtual methods i have a tiny little problem with xslt, js and html entities, eg. within a template:  &lt;script type="text/javascript"&gt;     &lt;xsl:value-of select="/some/node"/&gt;     for (var i = 0; i &amp;lt; 5; i++) {         //            ^^^ js error     } &lt;/script&gt;  &lt;script type="text/javascript"&gt;     &lt;xsl:value-of select="/some/node"/&gt;     for (var i = 0; i &lt; 5; i++) {         //            ^ xslt error     } &lt;/script&gt;  &lt;script type="text/javascript"&gt;     &lt;xsl:value-of select="/some/node"/&gt;     // &lt;![CDATA[     for (var i = 0; i &lt; 5; i++) {         //            ^ becomes &amp;lt;     }     // ]]&gt; &lt;/script&gt;   &lt;script type="text/javascript"&gt;     &lt;xsl:value-of select="/some/node"/&gt;     for (var i = 0; i &lt;xsl:value-of disable-output-escaping="yes" select="string('&amp;lt;')"/&gt; 5; i++) {         // works of course     } &lt;/script&gt;   does anyone have an idea where my problem could come from? i always thought the xslt processor would leave the content of a &lt;script/&gt; element unescaped when using the html output method ...  i run libxslt2 version 1.1.24 on OSX which was installed using macportsports ... Could $X^3 + X + 1$ have a root in $E?$ Consider the irreducible polynomial $g = X^4 + X + 1$ over $F_2$ and let $E$ be the extension of $F_2 =$ {0, 1} with a root $α$ of $g.$  Could $X^3 + X + 1$ have a root in $E?$  When working through this sheet Ive come across this question in which im not sure how to answer so any help will be appreciated How can I search for a keyword with special characters in Google Search? I want to search for what += means, but Google won't return any documents.  Also when I come across some PHP code I don't understand I can't search for the keywords as written.  How can I search for these special characters or include these special characters in a search. When your 10-year old boy says “It’s meta,” what does it mean? In what situation and of what sort of object they use this phrase? I asked about  a few days ago, quoting Maureen Dowd’s review of the movie, “J. Edgar” in New York Times.   I received six answers. But I still don’t get a clear idea of what “It’s meta” means because I don't understand (or have a total inability to comprehend) the concept of “self-referential.”   An answerer : “Meta in this fairly recent, casual context is supposed to mean self-referential, or recursive in some way. This is the sense in which my teenagers would use this term.”  So let me resubmit the question on “meta” in simpler format. When your teenager boy says “It’s (or this is) meta,” what does it mean? In what situation and of what sort of object they use this phrase?  I’m sorry for many users who lent me kind answers to my previous question. But I would like to get it fully on the meaning and usage of “it’s meta,” as a colloquial expression, not the meaning of meta as a prefix. I had internet connection, but when I upgraded my VGA and restarted my PC then everything for eth0 has gone.     When I used ifdown:     ifdown: interface eth0 not configured   Is there something I could do?    EDIT: Content of /etc/network/interfaces    auto lo iface lo inet loopback   When I ifconfig I don't have these lines:    inet addr:192.168.1.5  Bcast:192.168.1.255  Mask:255.255.255.0 inet6 addr: fe80::219:5bff:fe5e:a5e/64 Scope:Link   In the Terminal typed:  sudo gedit /etc/network/interfaces   and then added these lines in the bottom:  auto eth0 iface eth0 inet dhcp   then I typed:    sudo ifdown eth0 sudo ifup eth0   when I send the second command I received the below message:    No DHCPOFFERS received No working leases in persistent database - sleeping   Any help? What is the concept of erasure in generics in Java? What is the concept of erasure in generics in Java? , but how can I add or edit a URL (or URI) in the history? Proving a graph has no Hamiltonian cycle I have a DEM that I would like to smooth or generalize to remove topographic extremes (chop off peaks and fill-in valleys). Ideally, I would also like to have control over the radius or level of "blurriness". In the end, I will need a set of rasters that range from slightly blurry to really blurry. (Theoretically, the blurriest would be a constant raster of the arithmetic mean of all values).  Are there any tools or methods I can use (based on Esri, GDAL, GRASS)?   Do I need to home bake my own  routine?   Could I use a low-pass filter (e.g. ArcGIS's ), and if so, would I need to run it a bunch of times to get the effect of a large radius? Will airlines compensate me if my entertainment system is not working? In long haul flights, if I get a seat and after take off I found out that entertainment system is not working, I asked a flight attendant to change my seat but she apologized because there are no empty seats to move to.   Is it common to ask for a compensation for such an inconvenience? do major airlines compensate you in such cases? I am trying to execute a web service call to our backend system when the opportunity is closed/won. I am attempting to do a  POST to our endpoint. All the api call is really carrying the account name and a few structural detail back (All listed below). My problem is that when the trigger executes I get the following error in the debug logs "System.CalloutException: You have uncommitted work pending. Please commit or rollback before calling out".  The Process Builder Passing in the company name:   APEX Class  public class Play2 { @TestVisible static HttpResponse res;  @InvocableMethod public static void sendNotification(List&lt;String&gt; name) {        HttpRequest req = new HttpRequest();     Http http = new Http();     String payload = JSON.serialize(         new Map&lt;String, String&gt; {             'companyName' =&gt; name.get(0),             'salesforceAccountType' =&gt; 'Company',             'salesforceSynchStatus' =&gt; 'Enabled'         }     );      req.setEndpoint('https://TheEndPoint');     req.setMethod('POST');     req.setHeader('Authorization', 'The Mighty API KEY');     req.setHeader('Content-Type', 'application/json');      req.setBody( payload );     req.setCompressed(true);       try {         System.debug('Point 1');         res = http.send(req);         System.debug('Point 2');     } catch(System.CalloutException e) {         System.debug('Callout error: '+ e);         System.debug(res.toString());     } } } AMD GPU Material in "material preview mode" and "rendered mode", is entirely gray as background. Only outline is displayed How do I run two isolated instances of firefox? Is most of the human genome functionless “junk DNA”? In January 2012, David Icke published a book (ISBN: 9780955997334) entitled “Remember where you are and where you come from” in which he challenged what he claimed to be the view of mainstream science that 90–97% of the human genome was , serving no function. He implied, instead, that it must have a function, but that this was unknown.  Is it still true that we do not know what all this DNA does? I see that we consider this DNA to be non-coding, but does that mean it is “junk”? I find it hard to believe that it can serve no function, otherwise why would the DNA be there? Is it possible that it is used for things we don't yet understand? How can I safely prototype a mains power circuit? I'm building a device that has a mains power component, which I need to prototype before building to make sure everything works. I know I need to make sure wiring—and insulation!—is thick enough to cope with the higher current, and that I need to make sure there's enough space between exposed components so there's no arcing. I'll also be stashing whatever I prototype in a nice, insulated box before applying power, and I'll be include a small-value (250mA) fuse just in case. I'll be considering everything double-insulated rather than earthed (plastic enclosure, and there's nothing I'll need to touch while it's operating).  My plan is to solder (through-hole) components directly to each other, then insulate everything (with heatshrink tubing or electrical tape) as much as possible, then use screw-type cable joiners to connect the power cable.  I know I can't use traditional prototyping tools such as breadboards and stripboard for mains voltages, but what can I use to make prototyping at mains voltages easier?  (To be clear, 'mains voltage' in my area means 230V AC +10/-6% at 50Hz; GPOs are generally rated at 10A.)