Integral of the square of a non-standard gaussian pdf What is the integral of the square of a non-standard univariate gaussian distribution? The following is what I have so far, is it right? $\int_{-\infty}^{\infty} p^2(x)dx$, where $p(x)\sim N(\mu, \sigma^2)$ $\int_{-\infty}^{\infty} p^2(x)dx = \int_{-\infty}^{\infty} (\frac{1}{\sqrt{2\pi \sigma^2}} e^{\frac{-(X-\mu)^2}{2\sigma^2}})^2 dx$ $= \int_{-\infty}^{\infty} \frac{1}{2\pi \sigma^2} e^{\frac{-2(X-\mu)^2}{2\sigma^2}} dx$ $= \frac{1}{2\pi \sigma^2} \int_{-\infty}^{\infty} e^{\frac{-(X-\mu)^2}{\sigma^2}} dx$ $= \frac{\sigma \sqrt{\pi}}{2\pi \sigma^2} $ $= \frac{1}{2\sigma \sqrt{\pi}} $ And what is the integral if it is a 2D gaussian? $\int_{-\infty}^{\infty} p^2(x, y)dx$, where $p(x, y)\sim N(\mu, \Sigma^2)$ $\int_{-\infty}^{\infty} p^2(x, y)dx = \int_{-\infty}^{\infty} (\frac{1}{2\pi \sqrt{\Sigma}} e^{\frac{-(X-\mu)^2}{2\Sigma}})^2 dx$ $= \int_{-\infty}^{\infty} \frac{1}{4\pi^2 \Sigma} e^{\frac{-2(X-\mu)^2}{2\Sigma}} dx$ $= \frac{1}{4\pi^2 \Sigma} \int_{-\infty}^{\infty} e^{\frac{-(X-\mu)^2}{\Sigma}} dx$ $= \frac{\sqrt{\pi\Sigma}}{4\pi^2 \Sigma} $ $= \frac{1}{4\pi\sqrt{\pi} \sqrt{\Sigma}} $
Integral of the square of the normal distribution. Does someone know what the integral of the square of the normal distribution with standard deviation sigma over the range $-\infty$ to $+\infty$ is?  Thanks!
How do I customize the date/time display in GNOME Shell's top bar? Time indicator is shown as "%a %H:%M" in the center of the top bar.  How do I move it back to the right where it belongs, and make it include the full date and time ("%Y-%m-%d %s %H:%M")? I'm reading  and I'm trying to understand the proof under "Degeneracy in One Dimension".  Here's what it says:     Considering a one-dimensional quantum system in a potential $V(x)$ with degenerate states $|\psi_1\rangle$ and $|\psi_2\rangle$ corresponding to the same energy eigenvalue $E$, writing the time-independent Schrödinger equation for the system:       $$-\frac{\hbar^2}{2m}\frac{ \partial^2\psi_1}{ \partial x^2} + V\psi_1 =E\psi_1$$       $$-\frac{\hbar^2}{2m}\frac{ \partial^2\psi_2}{ \partial x^2} + V\psi_2 =E\psi_2$$   Multiplying the first equation by $\psi_2$ and the second by $\psi_1$ and subtracting one from the other, we get:       $$\psi_1\frac{d^2}{d x^2}\psi_2-\psi_2\frac{d^2}{d x^2}\psi_1=0$$   Integrating both sides       $$\psi_1\frac{\partial \psi_2}{\partial x}-\psi_2\frac{\partial \psi_1}{\partial x}=constant$$   In case of well-defined and normalizable wave functions, the above constant vanishes, provided both the wave functions vanish at at least one point, and we find: $\psi_1(x)=c\psi_2(x)$ where $c$ is, in general, a complex constant. So, the two degenerate wave functions are not linearly independent. For bound state eigenfunctions, the wave function approaches zero in the limit $x\to\infty$ or $x\to-\infty$, so that the above constant is zero and we have no degeneracy.   Here are the parts I don't understand:   How does integrating $\psi_1\frac{d^2}{d x^2}\psi_2-\psi_2\frac{d^2}{d x^2}\psi_1=0$ yield $\psi_1\frac{\partial \psi_2}{\partial x}-\psi_2\frac{\partial \psi_1}{\partial x}=constant$.  Without having explicit formulas for $\psi_1(x)$ and $\psi_2(x)$, how do we just apparently pull them both out of the integral? I don't understand how the constant in the last equation vanishes just because $\psi_1$ and $\psi_2$ vanish at infinity. Even if it does vanish, how does that imply $\psi_1=c\psi_2$? I am technically knowledgeable enough to know that this most likely isn't possible, BUT, I am curious just the same (it must be put to rest). Is there anything preventing us from creating a means to use two different internet connections, like two different ethernet cards or an ethernet card and a wireless card, and using the two IP addresses at once for more internet speed?  My specific situation is that I would like to use both a wireless card and my wired connection on my machine running Kubuntu 13.10. Using two internet connections would allow me to do things like:   Torrent and browse simultaneously without lag Open Firefox with multiple tabs faster (?) Download a couple or a few large files at once double time Increase overall internet connection speed (?) Debugging with command-line parameters in Visual Studio I'm developing a C++ command-line application in Visual Studio and need to debug it with command-line arguments. At the moment I just run the generated EXE file with the arguments I need (like this program.exe -file.txt) , but this way I can't debug. Is there somewhere I can specify the arguments for debugging? I have the following enumeration:  public enum AuthenticationMethod {     FORMS = 1,     WINDOWSAUTHENTICATION = 2,     SINGLESIGNON = 3 }   The problem however is that I need the word "FORMS" when I ask for AuthenticationMethod.FORMS and not the id 1.  I have found the following solution for this problem ():  First I need to create a custom attribute called "StringValue":  public class StringValue : System.Attribute {     private readonly string _value;      public StringValue(string value)     {         _value = value;     }      public string Value     {         get { return _value; }     }  }   Then I can add this attribute to my enumerator:  public enum AuthenticationMethod {     [StringValue("FORMS")]     FORMS = 1,     [StringValue("WINDOWS")]     WINDOWSAUTHENTICATION = 2,     [StringValue("SSO")]     SINGLESIGNON = 3 }   And of course I need something to retrieve that StringValue:  public static class StringEnum {     public static string GetStringValue(Enum value)     {         string output = null;         Type type = value.GetType();          //Check first in our cached results...          //Look for our 'StringValueAttribute'           //in the field's custom attributes          FieldInfo fi = type.GetField(value.ToString());         StringValue[] attrs =            fi.GetCustomAttributes(typeof(StringValue),                                    false) as StringValue[];         if (attrs.Length &gt; 0)         {             output = attrs[0].Value;         }          return output;     } }   Good now I've got the tools to get a string value for an enumerator. I can then use it like this:  string valueOfAuthenticationMethod = StringEnum.GetStringValue(AuthenticationMethod.FORMS);   Okay now all of these work like a charm but I find it a whole lot of work. I was wondering if there is a better solution for this.  I also tried something with a dictionary and static properties but that wasn't better either. Is it required for panel data to use dummy variables? I am doing research considering seven countries and I have panel data. My question is: do I need to include dummy variables every time I use panel data in regression, or is enough to do it as a fixed effect in Stata, without dummy variables? Ordered N-Tuples as Cartesian Products of Indexed Families In Introduction to Topology, Mendelson defines the Cartesian Product of an Indexed Family as follows:     DEFINITION  Let $\{X_\alpha\}_{\alpha \in I}$ be an indexed family of sets. The product  of the sets $\{X_\alpha\}_{\alpha \in I}$,    written  $\prod_{x\in I}X_\alpha$ consists of all functions $x$ with domain the indexing set $I$ having the property that for each $\alpha \in I$, $x(\alpha)\in X_\alpha$.    Halmos, in Naive Set Theory, makes the following observation about this sort of product:     Ordered triples, ordered quadruples, etc., may be defined as families whose index sets are unordered triples, quadruples, etc.   However, it is not immediately apparent to me why this is true. It makes sense if we understand the prior assertions in Halmos that $\prod_{i \in I} X_i = X_i$ and that $\prod_{i \in I} X_i = X_a \times X_b$ when $I = \{a, b\}$, but these already pose difficulty for me.  Let $I = \{0, 1\}$, let $X_0 = \{A, B\}$, and let $X_1 = \{C, D\}$. Then by Mendelson's (and Halmos's) definitions, it would seem to me that . . .  $$\prod_{i \in I} X_ i = \{\{(0, A), (1, C)\}, \{(0, A), (1, D)\}, \{(0, B), (1, C)\}, \{(0, B), (1, D)\}\}$$  This would appear to me to be the set of all functions matching the definition given, but it is not equal to $X_0 \times X_1$. Moreover, extending this to a case where the index set $I$ is a triple, we would not end up with ordered triples, but with each subset of the product containing three ordered pairs.  When Halmos posits the equalities he does, the idea seems easy to grasp, but the peculiar wording of Mendelson's definition suggests something different to me. I have a directory that I am trying to clean out that contains both files and subdirectories.  What I am trying to do is simple: move all the files into another directory, but leave all the sub-directories as they are.  I am thinking something like:  mv [*_but_no_dirs] ./other_directory   It seems like there should be a simple way to do this with wildcards * and regex...  Anyone have ideas? How to pack the game files into one without using any file compressor? Well, one example is this file of the game The Sims 2 for PS2: quickdat.arc. The file contains this .arc extension, but it's no one an archive file and it contains some data of the game inside. How can I do this? Put so many files in one?  How to do something like this? All the game files inside one without using a file compressor? I want to ask about two sentences, and what would be the correct way to form a question:   It's very important that we eat healthy food, ...........?  Which would be the correct question tag: (isn't it) or (don't we)? It's a must that all applicants follow the rules of the competition, ..........?  Which would the correct question tag: (don't they) or (mustn't they)?   Please could you tell me the correct answers, and, more importantly, why those are correct? Is there a single word for “one who explains his point of view very clearly”? Cross-platform space remaining on volume using python I need a way to determine the space remaining on a disk volume using python on linux, Windows and OS X. I'm currently parsing the output of the various system calls (df, dir) to accomplish this - is there a better way? My Area51 account crossed 200 rep today, but still other accounts don't reflect the change in reputation. When will the changes be applied? or is there no association bonus for Area51 account? Is it possible to tell when statistics were last accessed by a query using T-SQL? I'm trying to ascertain what statistics are involved in a query, I know I can check the visual query plan but I wondered whether there was a table (or tables) with this information. Every closed subset $E\subseteq \mathbb{R}^n$ is the zero point set of a smooth function How to Async.AwaitTask on plain Task (not Task)? For realism should I use the maximum number of Subsurf levels? I was wondering if I always need to use the maximum value of the options from the elements of "Add Modifier". For example, do I need to use the view 6 in Subdivision surface? I know it depends of what we are trying to achieve, but we all want the same. Make something look the most believable. So do you always use the maximum value Blender offers? Social network login or not? I am developing a niche social networking iPhone app of my own.   I was wondering if it would be best to allow the user to connect to social networks such as Facebook and Twitter to login to my app (and of course an eMail/Password fallback) or just go with regular eMail and Password login.  Any thoughts would be appreciated! :) Behaviour of increment and decrement operators in Python Some people have a diamond after their username (e.g. )    What special privileges do diamond moderators have? How can I become a diamond moderator? Who are the diamond moderators?  How many are there?      For more information, see  in the .     Note: not to confuse with certain Stack Exchange employees, for example  members or "community managers", who often have diamonds as well, but are not ordinary moderators.