In this post: , I am having difficulty understanding some of the steps. We have : (a) $s^2=\frac{1}{n-1}\sum_{i=1}^{\infty}(x_i-x\bar)^2$ and further down, (b) $E(s)= \sqrt\frac{\sigma^2}{n-1} E(\sqrt\frac{s^2(n-1)}{\sigma^2})$ I understand the 2nd and 3rd lines, but how do we get from (a) to (b)? (My knowledge of maths is basic)
According to the Wikipedia article on  the sample SD   $$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2}$$  is a biased estimator of the SD of the population. It states that $E(\sqrt{s^2}) \neq \sqrt{E(s^2)}$.   NB. Random variables are independent and each $x_{i} \sim N(\mu,\sigma^{2})$   My question is two-fold:   What is the proof of the biasedness? How does one compute the expectation of the sample standard deviation    My knowledge of maths/stats is only intermediate.
How to add a program as a startup application from a terminal? I am using Ubuntu 12.04 and I know the GUI way to maintain the startup program list but now I want to find out the way to add/remove startup applications using a command line interface (CLI). Create default home directory for existing user in terminal How do I do this? I created a user without a home directory and now I want to create one. Not just a folder called new_user, I want the default home directory with all it's folders and hidden files, etc. My Developer Account cannot be activated because I used a relative's credit card Given $n+1$ points in the plane, Lagrangian interpolation gives the unique polynomial of degree less or equal to $n$.  From the examples I've seen so far, interpolation polynomials all have degree exactly $n$ but I don't know if there might be a set of points for which the interpolation gives a polynomial of degree strictly less than $n$.  Many thanks for your answers. Why does the connection time out when connecting to a server? My significant other and I finish our master degrees at the same time and thus we are applying for PhD programmes at the same time. One major factor of accepting a potential offer would be if we could study at the same university, so I feel that mentioning a two-body problem would be advantageous for both, us and the admission committee. But where do I mention this? Is a line in the statement of purpose suitable here?     If you admit me, please also consider admitting my significant other Jane Roe.   Some additional information: We both do research in computer science. While I do research in natural language processing my partner does research in computer vision. Hosting multiple distinct folders for distinct domains I have a VPS that I've installed a Apache Web Server on. What I want to do is forward my own website domains to that server and host those websites' files inside my Apache Web Server. As far as I understand a DNS server is required to manage which domain is hosted where (IP-wise). But what I want to do is host my websites under the same IP address, the one that my web server has. How can I separate those websites' files inside my web server so I don't have to use http://host/foo for www.foo.com and http://host/bar for www.bar.com? Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$? Call to a member function on a non-object could you help me on this integral ? $$\int \frac{1}{\cos(x)}\,\mathrm dx$$ Here's what I've started : $$\int \frac{1}{\cos(x)}\,\mathrm dx = \int \frac{\cos(x)}{\cos(x)^2}\,\mathrm dx = \int \frac{\cos(x)}{1-\sin(x)^2}\,\mathrm dx$$ Now, I did : $u = \sin(x)$, so $\mathrm du = 1$. Now I have : $$\int \frac{\text{???}}{1-u^2}\,\mathrm du$$ But at this point, I think I did the most of the job but I'm stuck. Could you help me to solve this integral please (to the integration by substitution at the end) ? Thanks EDIT : Now I follow the steps and I got : $$\int \frac{1}{1-u^2}\,\mathrm du$$ Doing the partial fraction I got $A = 1/2$ and $B = 1/2$. So basically I have : \begin{align} &amp; \int \frac{1}{1-u^2}\,\mathrm du = \int \frac{1/2}{1+u}\,\mathrm du + \int \frac{1/2}{1-u}\,\mathrm du \\[8pt] = {} &amp; \frac 1 2 \left(\int \frac{1}{1+u} \, du - \int \frac{1}{1-u} \, du\right) \\[8pt] = {} &amp; \frac 1 2 \ln\left(\frac{1+u}{1-u}\right) = \ln\left(\left(\frac{1+\sin(x)}{1-\sin(x)}\right)^{1/2}\right) \\[8pt] = {} &amp; \ln \left(\frac{\sqrt{1+\sin(x)}}{\sqrt{1-\sin(x)}}\right) = \ln\left(\frac{\sqrt{1+\sin(x)}}{\sqrt{\cos(x)^2}}\right) \\[8pt] = {} &amp; \ln\left(\frac{\sqrt{1+\sin(x)}}{\cos(x)}\right) \end{align} Here's what my teacher got :  What's wrong with what I did ? Did I miss something ? Supremum and infimum of sets I've recently started to study supremum and infimum, and I'm having trouble determining the supremum and infimum of the following sets:  $ A = \left\{ {mn\over 1+ m+n} \mid m, n \in \mathbb N \right\} $ $ B = \left\{ {mn\over 4m^2+m+n^2} \mid m, n \in \mathbb N \right\} $ $ C = \left\{ {m\over \vert m\vert +n} \mid m \in \mathbb Z, n \in \mathbb N \right\} $ $ D = \left\{ {n\over 3} - \left[{n\over 3}\right] \mid m, n \in \mathbb N \right\} $   In $D$, the $[\; ]$ denotes the integral part of ${n\over 3}$.   $E = \{x \mid x \text{ is decimal fraction between } 0 \text{ and } 1 \text{ that has the digits } 0 \text{ and } 1\}$.  In my progress so far I was able to verify that sup $\{x \in A\} = +∞$  Setting $m = (n-1)$ to get  $$ \frac{mn}{1+m+n}= \frac{n(n-1)}{1 + n + n-1} = \frac{n(n-1)}{2n} = \frac{n-1}{2} $$  That way $\frac{n-1}{2}\in A$ for every $n\geq 2$. Thus, the supremum of $A$ is bigger than $\frac{n-1}{2}$ for every integer $n$.  I don't know how to go past this. x_x  Any help would be greatly appreciated! What command can I type into the Terminal so that I can delete all .svn folders within a folder (and from all subdirectories) but not delete anything else? How do I record a screen cap video of the iPad? I'm looking for a method to record the video without actually filming the device. Does such a method exist? Normal text is invisible when using beamer with notes and XeLaTeX Activity has leaked window com.android.internal.policy.impl.PhoneWindow$DecorView@44f72ff0 that was originally added here Can Barbarians use 2 two-handed weapons in Diablo 3? Can barbarians dual wield two-handed weapons?  My barbarian character is only a level 50 now, but I'm not sure if I can dual wield two-handed weapons at a later level or not - or is there a special skill I need?  UPDATE It is sweet, that Blizzard have finally updated Diablo 3, so that the Crusader can have a 2-handed weapon in 1 hand ( passive). It is a shame that the Crusader can't dual wield a 2-handed weapon in both hands. Prove that a finite union of closed sets is also closed Let $X$ be a metric space.  If $F_i \subset X$ is closed for $1 \leq i \leq n$, prove that $\bigcup_{i=1}^n F_i$ is also closed.  I'm looking for a direct proof of this theorem.  (I already know a proof which first shows that a finite intersection of open sets is also open, and then applies De Morgan's law and the theorem "the complement of an open set is closed.")  Note that the theorem is not necessarily true for an infinite collection of closed $\{F_\alpha\}$.  Here are the definitions I'm using:  Let $X$ be a metric space with distance function $d(p, q)$.  For any $p \in X$, the neighborhood $N_r(p)$ is the set $\{x \in X \,|\, d(p, x) &lt; r\}$.  Any $p \in X$ is a limit point of $E$ if $\forall r &gt; 0$, $N_r(p) \cap E \neq \{p\}$ and $\neq \emptyset$.  Any subset $E$ of $X$ is closed if it contains all of its limit points. How do I remove the OneDrive folder from my File Explorer folder tree in Windows 10? I don't use or want to use OneDrive, but after upgrading to Windows 10 it's been installed automatically and added to the sidebar of every explorer window:    I can't seem to figure out how to get rid of it. I can't uninstall it like a normal app, and I can't seem to unpin it in Explorer's options.  I tried following  to getting rid of it, but that seems to only remove stored OneDrive files.  I've also tried following another  and , but I can't seem to edit group policies on my computer (maybe because I upgraded from Win7 Home Premium?).  All I would really like is to be able to remove the OneDrive shortcut from my explorer sidebar, though being able to uninstall OneDrive would be a bonus.  I'm not sure what I should try now. Is there a way to remove OneDrive from Windows 10? Another question on almost sure and convergence in probability Convergence in probability implies convergence on a subsequence almost surely.  But this means we fix a subsequence, such that $X_{n_k}$ converges for almost every $\omega$, right? The subsequence we pick does not depend on the $\omega$ right? Routing in home LAN I'm trying to understand some basic concepts of networking. I have following home LAN:     Is it right (like above) to have 2 NIC's each one having 2 different gateways on one station? There PC2 have 2 NICs - the first's one gateway is 192.168.0.1 the second's 192.168.1.1? Initially (just after system startup) the ip table is like that:  192.168.0.0    255.255.255.0         On-link      192.168.0.11    276 192.168.0.11   255.255.255.255       On-link      192.168.0.11    276 192.168.0.255  255.255.255.255       On-link      192.168.0.11    276    and when i run netcat:  nc -zv -s 192.168.0.11 192.168.0.11 80    to test if port 80 is open it works - the port 80 is indeed open. But when i remove the 192.168.0.11 route i got "TIMEOUT" on nc test. Shouldn't it pick 192.168.0.0 route and work? Moreover when i add the route manually:   route add 192.168.0.11 mask 255.255.255.255 0.0.0.0   it still doesn't work! I can't change the metric to desired value. When i run for eg. route change 192.168.0.11 mask 255.255.255.255 192.168.0.11 metric 500 the result metric is't 500 but 520! (no matter if auto-metric is enabled or not)? Is there any connection between -s flag (local source address) in netcat and routing table's Interface column?