If $X\sim\mathcal{LN}({\mu,\sigma^2})$, then $\mathrm{E}[X]=e^{\mu+\sigma^2/2}$.  My question is: what right do we have to add a mean and variance together?  If $X$ has physical dimensions, then the expression $\mu + \sigma^2/2$ is incoherent.  So what gives?
I have a distribution of microparticles that follows a lognormal distribution. The cumulative distribution function thus is given by:  $$ F_X(x;\mu,\sigma) = \frac12 \operatorname{erfc}\!\left(-\frac{\ln x - \mu}{\sigma\sqrt{2}}\right) $$ $$ \mu = ln(M) + \sigma^2 $$  Now, the plot of the distribution function should be exactly the same no matter if the particle diameter $x$ is given in micrometers or meters (as long as I adapt the x-axis accordingly of course). However, this only works if I only convert $x$ and $M$, while not touching the numerical value of $\sigma$, and I don't understand why. $F_X$ has to be unitless, so $x$, $\mu$ and $\sigma$ should all have the same unit, right?
How to remove extra start item from boot menu? I bought a laptop recently. While Windows 7 was installing, the battery died. So I plugged in the charger and re-installed Windows 7 Home Basic from the start. After completion of the procedure I now have 2 Windows folders in my C:\ drive, so I deleted the old Windows folder.  The problem is, while booting time it shows two Windows 7 options, although I already deleted the old Windows folder.  How can I remove old Windows 7 option from my boot? SQL JOIN - WHERE clause vs. ON clause After reading it, this is not a duplicate of . The answer may be related (or even the same) but the question is different.    What is the difference and what should go in each?  If I understand the theory correctly, the query optimizer should be able to use both interchangeably. How can I identify what application is using a given file? I'm trying to delete some folder but this error is preventing from doing it:     Cannot delete [FileName]: It is being used by another person or program.      Close any programs that might be using the file and try again.   How can I identify what application is using the file?   I want to end the application using a Task Manager, but I didn't know what application is using it.  The folder is not shared and my user account is an administrator. After doing a sudo apt-get install , where does the application get stored to? After installing apps via apt-get install -app name-. Where does the files get copied to? Are the copied to just one location or multiple locations? And if so how do I find this out?  Here's an example: Changing deleted partition back to NTFS without having to recover to another drive Recently, I accidentally deleted a partition off of my external hard disk. I had a copy of Minitool Power Data Recovery left over when the same drive had become RAW about a year ago. When I selected lost partition recovery, it immediately identified my deleted partition and allowed me to access and recover the files that were there.  My question is this. Is there any way to change the deleted partition back to NTFS without having to have a drive that you need to recover the files to and then move them back after formatting, as I don’t have any space left on either the computer or on my other external hard drive, so that I don’t need to buy a new one. The files on the disk were really important.  I am using Windows 7 Home Premium.  I appreciate any help in solving this crisis. Time travel film: character with male and female reproductive organs is taken back in time by a barman who turns out to be themselves I have had a friend give me a rough description of a film and I'm curious to find out what it actually is. I don't know very much. I understand this will be difficult, especially since every time my friend explains the tale it's different so this description is likely to contain some inaccuracies.  The film's plot begins with a recounting given by the main character, who had both male and female reproductive organs but had one set removed (not sure which, think male ones were removed). The main character is narrating this tale to the barman in a bar. The barman then takes the main character back in time, where they make love to themselves and produce a child.  The child, once born, is abandoned by the mother in the past - but the barman takes the child to the orphanage and leaves them there, ensuring their survival. Eventually it is revealed that the child was the main character themselves, and so was the barman.  In other words every major character introduced up to that point was the same person who had severely messed with their own timeline. Do redirection operators always open file descriptors in parallel? 1. Consider snippet#1:  $ cat test.txt &gt; test.txt cat: test.txt: input file is output file   It seems that cat makes its input file descriptor point to test.txt and then when it tries to set its output file descriptor to test.txt it throws the above error. Here it seems that cat is aware of the redirect operator and so proceeds to attempt to set the output file descriptor to test.txt  2. Consider snippet#2:  $ cat 1.txt 1:CAT 2:dog $ sed 's/cat/CAT/g' test.txt 1:CAT 2:dog $ sed 's/cat/CAT/g' test.txt &gt; test.txt $ cat test.txt # Note that test.txt is now empty $   Here we see that sed opens test.txt (last argument) in read mode and at the same time sets test.txt as its file output descriptor. Also the '&gt;' operator overwrites the contents of the file BEFORE sed starts to read from it.   I am aware that commands in a pipeline execute in parallel but have not come across any info on how redirection operators behave. Any supporting links would be helpful. A "non-believer" of C# was asking me what the purpose to extension methods was. I explained that you could then add new methods to objects that were already defined, especially when you don't own/control the source to the original object.  He brought up "Why not just add a method to your own class?" We've been going round and round (in a good way). My general response is that it is another tool in the toolbelt, and his response is it is a useless waste of a tool... but I thought I'd get a more "enlightened" answer.  What are some scenarios that you've used extension methods that you couldn't have (or shouldn't have) used a method added on to your own class? What is the center of $D_{2n}/Z(D_{2n})$ What is the center of $D_{2n}/Z(D_{2n})$. I see that when $n=2^k$ then I have a $p$-group so the center is not trivial. But when $n$ is not power of $2$ how can i know what is the center of this group?. Surely when $n$ is odd then the center is trivial. The reminder email says:     We noticed you have not accepted our invitation to join Stack Overflow Careers. Only 3 days remain before it expires [...]      But please act soon: this invitation expires in 14 days [...]   Is it 3 days or 14 days now? (used to show two posts from MSE)   (there used to be a list of the accounts)   (it was there) seems to be fixed I was just wondering about that because it would influence how I deal with closed questions, e.g. whether they need to be improved (tagging, formatting, code samples) after closing, or if some should be deleted altogether.  However, I'm not asking for answers to these questions, I just want to know if Google (and/or other search engines) list closed questions in their search results. (I hope they don't.) Supremum of a product measurable function... I understand test -n &lt;expression&gt; to return false if the expression evaluates to a string length of greater than 0.  Why then does the following happen?  Macbook:~ echo ${#undeclared_variable} 0 Macbook:~ test -n $undeclared_variable &amp;&amp; echo Foo Foo   Bash 4.x  My expectation is that since test -n $undeclared_variable should evaluate false, that Foo should not be echoed.  If I use [[  ]] instead of test, it works as expected. What is "Best Practice" For Comparing Two Instances of a Reference Type? I came across this recently, up until now I have been happily overriding the equality operator (==) and/or Equals method in order to see if two references types actually contained the same data (i.e. two different instances that look the same). I have been using this even more since I have been getting more in to automated testing (comparing reference/expected data against that returned). While looking over some of the  I came across an  that advises against it. Now I understand why the article is saying this (because they are not the same instance) but it does not answer the question:  What is the best way to compare two reference types? Should we implement ? (I have also seen mention that this should be reserved for value types only). Is there some interface I don't know about? Should we just roll our own?!  Many Thanks ^_^ Update Looks like I had mis-read some of the documentation (it's been a long day) and overriding  may be the way to go..  If you are implementing reference types, you should consider overriding the Equals method on a reference type if your type looks like a base type such as a Point, String, BigNumber, and so on. Most reference types should not overload the equality operator, even if they override Equals. However, if you are implementing a reference type that is intended to have value semantics, such as a complex number type, you should override the equality operator. Tikzcd conflict with babel and cd I have this problem: when I try to compile a diagram I get the following message:     I think the culprit is a tikzcd arrow in ell 1-1   But I have not problem when, in the tikzlibrary, I just use cd, matrix and backgrounds. The problem is when using babel and cd at the same time.  \documentclass{article} \usepackage{amsfonts,amssymb,amsmath,tikz} \usepackage[spanish]{babel} \usepackage[latin1]{inputenc} \usepackage[mathcal]{euscript}  \usetikzlibrary{babel,cd,matrix,backgrounds}  \begin{document} \begin{center}   \begin{tikzcd}     X\rar[mapsto]{T}&amp;Y   \end{tikzcd} \end{center}  \end{document} $$\lim\limits_{x\to 1}\frac{\sqrt x-x^2 }{1-\sqrt x}$$  I'm not entirely sure how to go about this, since rationalizing anything doesn't quite work for me. Just looking for some help. What GUI applications are there to assist in generating graphics for TeX? Design Review: 3.3V Logic to 12V Logic (Uni-directional and Bi-directional) How to make the tick thickness as the axis line? I am trying to make the thickness of the ticks equal to the axis line thickness. That is what I have so far:  \documentclass{article}   \usepackage{pgfplots}  \pgfplotsset{compat=1.5}   \begin{document}     \begin{tikzpicture}     \begin{axis}[         xlabel=Binding Energy (eV),         ylabel=Intensity (a.\,u.),         xtick pos=left,         x dir= reverse,         ytick = \empty,         every x tick/.style={color=black, thin},         tick align=outside,         xlabel near ticks,         ylabel near ticks,         xmin = 96,         xmax = 108     ]         %\addplot[mark=none] table[x=BE,y=cnts] {si2p.txt};         % This data looks similar:         \addplot[mark=none, smooth] table[x=BE,y=cnts] {             BE      cnts             95      1             98      1             99      10             99.5    5             100 6             101 2             108 2         };      \end{axis}     \end{tikzpicture} \end{document}     But if you take a closer look on the ticks on the outside, you'll see that they are somehow clipped or a least thinner. They are also black so the style seems to work for them.    Is this a bug or can I change the clipping region? And as a second question: is thin the correct value for the thickness?