I am trying to find $\mathbb{E}(e^{-X})$ where $X$ is a random variable with a general normal distribution. I end up with  $$(2\pi \sigma)^{-\frac{1}{2}} \int_{-\infty}^{\infty} e^{-x}e^{\left(\frac{-(\mu-x)^2}{2\sigma^2}\right)}dx $$  Tried some transformations and the polar coordinate trick used to derive values of $(2\pi \sigma)^{-\frac{1}{2}}\int e^{(\frac{-(\mu-x)^2}{2\sigma^2})}dx $ but haven't been able to come up with anything explicit. Can anyone point me in the right direction? I am in despair... do I need to use a numerical method?   If it helps, here is what the integral boils down to $$(2\pi \sigma)^{-\frac{1}{2}}\int \exp\left(-\frac{x(2\sigma - 2\mu +x)}{2\sigma^2}\right)dx $$
I'm working through some old integrals and I found one that's interesting. I can't quite remember how it's proved, so if someone could set me off in the right direction, it would be really helpful. Thanks!   $\int_{\mathbb{R}} e^{-ax^2 + bx} dx = \sqrt{ \dfrac{\pi}{a}} e^{\frac{b^2}{4a}}$  I've tried change of variable, but I'm not sure this is the right approach.
Calculate mean of ordinal variable I've read in a number of places that calculating the mean of an ordinal variable is inappropriate. I'm trying to get an intuition for why it might be inappropriate. I think it is because, in general, an ordinal variable is not normally distributed and so calculating the mean will give an inaccurate representation. Could somebody give more detailed reasoning for why calculating the mean of an ordinal variable might be inappropriate? Why redirect output to 2>&1 and 1>&2? I have come across several commands that use 2&gt;&amp;1 and 1&gt;&amp;2, but I quite can't get my head around the purpose of using it and when I should be using it.  What I understand  I know that 1 represents standard out and 2 represents standard error. I understand that 2&gt;&amp;1 combines the output of 2 to 1 and vice versa.  What I don't get   When should I be using it? What purpose does it serve? How to relate areas of circle, square, rectangle and triangle if they have same perimeter?? I was given a question which was like:     Suppose that a circle, square, rectangle and triangle have same perimeter. How area their areas related??   My work:  I broke the question in parts and tried to prove it seperately:  STEP $1$. Suppose the given perimeter is $P$.So the radius of circle will be ${r=P/2\pi}$.  And hence area of the circle will be:   ${P^2/4\pi}$.  Side of square will be $P/4$ and hence area of square will be $P^2/16$.   It is obvious that ${P^2/4\pi}$  >  $P^2/16$.   So Area of circle is more than area of square.  STEP $2$: Whether by using $AM&gt;GM$ inequality or by using a bit of differentiation, we can get that product off two quantities is maximum when they are equal.  So, Area of square is more than Area of rectangle.  Here, the hardest part come (at least for me). I can't relate the area of triangle to any of Circle, Square or Rectangle. It would have been easy if the triangle was Equilateral, Right angle but the question is talking about a general triangle.  However, when I considered the situation with different examples (Assuming the perimeter and calculating the area, I got the relation:  Circle > Square > Rectangle > Triangle.  I shall be highly thankful if you guys can establish the relation in purely mathematical way.  EDIT: This question does not ask that whether circle have the largest area among all 2-D figures or not. The OP just want to relate the areas of Circle, Square, Rectangle, Triangle in a pure mathematical way. I remember reading a series when I was around 7th or 8th grade about aliens who had taken over Earth by putting metal caps on people when they were "entering adulthood" to control them (not direct control, but it would keep them from rebelling). There was a faction of people who able to escape and had fake caps and would save teens who were about to be capped.  The main protagonist was a teen boy who was a boxer (or training to be a boxer or something). The aliens would hold tournaments for the humans and whoever won would be "worthy" to serve as personal slaves to the aliens.     The protagonist won and was able to get inside to try and learn more about the aliens from the inside. How can I import the configuration on a different site? I have this error when I try to import the config of an another site:     The staged configuration cannot be imported, because it originates   from a different site than this site. You can only synchronize   configuration between cloned instances of this site.   I have 10 customers, and I need to sync my site configuration with 10 different websites. The site UUID is a problem for sharing a configuration with a different website. Is there a solution? How do I install Ubuntu alongside a pre-installed Windows with UEFI? I'm absolutely new to Linux. I would like to know how to install Ubuntu alongside the pre-installed Windows 8+ OS.  Should I do it with Wubi, or through the Live USB/DVD?  What steps do I need to take to correctly install Ubuntu? What should I do if I have two libraries that provide functions with equivalent names? I have been denied a visa several times in a row. How does one deal with such serial refusals? Over the course of time here on TSE, I have observed individuals whose pattern of visa applications is best described as ''...     serial refusals - Two or more consecutive visa refusals issued by the same regime to the same person. Commonly for the same, or nearly the same, reasons. In some visa regimes Serial refusals can have a cumulative effect wherein the applicant's credibility is increasingly eroded, especially when the application pattern suggests a secondary agenda.   A common thread appears to be the need to visit the destination country (especially Schengen, the USA, or the UK) with a special sense of urgency. Also common in these cases are the following types of questions...   Should I just reapply once again? What are my chances? What can I do to improve my odds in the future? Can I somehow address the previous refusals to avoid them being a burden on all future applications? How can I break the pattern?      NB: This question is intended as a canonical about serial refusals. I have never been personally denied a visa. Where did the name "Stack Exchange" come from? I get Stack Overflow (which I'm guessing was the first site?) and Super User, Code Review, and the other sibling site names make sense to me.  But I've never heard of a stack exchange before.  Is a stack exchange a real thing, or is this site's name a combination of Stack Overflow and an exchange of ideas? Showing $\gcd(2^m-1,2^n+1)=1$ Is RSA key size the size of private key exponent? I have implemented a key pair generation scheme for RSA algorithm. I have taken the length of private key exponent as RSA key size, but then I've got to know that RSA key size is the size of the modulus $n$. Is it wrong to take RSA key size as the size of the private key exponent?  To generate a private key of 1024 bits, I increase the size of private key exponent regardless the modulus $n$. Cannot see Samba configuration tool in the Dash in ubuntu 14.04 After a fresh installation of ubuntu 14.04 (my previous installation has seen to be broken somewhere reading .wmv file with VLC), I've installed Samba GUI tool from the software center.  It briefly appears in the launchpad then disappears. From this time I cannot the see in in the dash.  I've found this in askUbuntu   but the command sudo apt-get install samba samba-common gives me nothing to install.  Lecture des listes de paquets... Fait Construction de l'arbre des dépendances        Lecture des informations d'état... Fait samba est déjà la plus récente version disponible. samba passé en « installé manuellement ». samba-common est déjà la plus récente version disponible. Les paquets suivants ont été installés automatiquement et ne sont plus nécessaires :   kde-l10n-engb kde-l10n-fr Veuillez utiliser « apt-get autoremove » pour les supprimer. 0 mis à jour, 0 nouvellement installés, 0 à enlever et 8 non mis à jour.   The following guide  has no result either.  Any idea ?  Laurent06000 Say $X_1, X_2, \ldots, X_n$ are independent and identically distributed uniform random variables on the interval $(0,1)$.  What is the product distribution of two of such random variables, e.g., $Z_2 = X_1 \cdot X_2$?  What if there are 3; $Z_3 = X_1 \cdot X_2 \cdot X_3$?  What if there are $n$ of such uniform variables? $Z_n = X_1 \cdot X_2 \cdot \ldots \cdot X_n$? How to `bundle install` when your Gemfile requires an older version of bundler? A cyclic group is a group that is generated by a single element.  That means that there exists an element $g$, say, such that every other element of the group can be written as a power of $g$.  This element $g$ is the generator of the group.  Is that a correct explanation for what a cyclic group and a generator are? How can we find the generator of a cyclic group and how can we say how many generators should there be? Old post apocalyptic cartoon where the main character has red hair and lives with a nomadic group Removing unwanted appearance of underlying mesh Software to synchronize two directories (local/remote) I need software that would detect changes in directory (A) every 5 seconds or so and upload/delete changed files from/at (B) remote directory.  I found non-commercial software, . I am wondering if there are any commercial alternatives? Unable to uninstall Wine apps I recently ran the iTunes 10 installer in Wine and everything seemed to go well. Then, when I tried to run it, it would constantly crash/and the appearance would be glitchy. After looking at the Wine website, it turns out that iTunes 10 has a 'garbage' rating of Wine compatability.  Now I'm trying to remove/uninstall iTunes, QuickTime, and the other applications that come default with installing iTunes (such as Bonjour, etc.). However, when I run the 'uninstaller' that wine provides for the programs, it instead executes what appears to actually be the original installer! I've tried deleting all the configuration/data files in my Wine's file-system, but to no avail.  Any help is greatly appreciated!  Edit1: After re-running the uninstall process and then rebooting, the program (iTunes 10) still remained installed. Proving the series $\sum_{n=0}^{\infty}\sin(e\pi n!)$ converges