A friend of mine is working analysing 2000 twits per day and categorize them as postive, negative or neutral.  This is a really boring task but the algorithms that do this classification are not very good because they can't detect sarcasm. A simple solution to make more easy this task is to do a subsample of the original $N = 2000$ data points.   Doing some tests we saw that with $30\%$ of the data the normalized histograms of the subsample and the original data points look very similar but we need to know a better estimation of the error of doing this subsample.  Theoretically the data points are an i.i.d. sequence $(X_i)_{i=1}^N$ (big assumption) in the space $A = \{0,1,2\}$ (positive,negative,neutral). Let $(X_{(i)})_{i=1}^n$ be a subsample  of size $n \leq N$  (draw $n$ elements uniformly without replacement). In some sense I want to characterize the distribution of $(X_{(i)})_{i=1}^n$ in order to choose a $n$ such that the empirical distribution of $(X_{(i)})_{i=1}^n$  is close to the empirical distribution of $(X_i)_{i=1}^N$.   Any help will be appreciated
I have a population of phone calls - 200,000.  There are different reasons for each call, but lets assume the number of reasons is known. i.e. 7 different call reasons: 1) Check on order 2) Cancel order 3) Billing information 4) Account information etc...  My question is, I want to get a statistically significant sample size so the number of calls I listen to in my sample are representative of the distribution of the reasons in the overall population of 200,000 calls.  What should the sample size be? Which methods to use to calculate the sample size?
My csv export displaying html, how to get rid of? How can blackbody radition be explained by quantization? I don't understand why quantization makes a peak on the blackbody radiation curve (so there is no ) and the relationship between that peak and quantization concept.  When the blackbody is heated, it starts to glow. All atoms start to vibrate. Total heating energy must be divided and equally shared by all atoms of blackbody. Right?  But what happens next?  Do some atoms get much more energy to radiate at higher frequency and waste the energy? What happens at the atomic level, so quantization can explain it? What does classical physics claim, so it can not explain the radiation curve? Motivation behind standard deviation? Let's take the numbers 0-10.  Their mean is 5, and the individual deviations from 5 are -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 And so the average (magnitude of) deviation from the mean is $30/11 \approx 2.72$.  However, this is not the standard deviation.  The standard deviation is $\sqrt{10} \approx 3.16$.  The first mean-deviation is a simpler and by far more intuitive definition of the "standard-deviation", so I'm sure it's the first definition statisticians worked with.  However, for some reason they decided to adopt the second definition instead.  What is the reasoning behind that decision? Environment for two columns with lstlistings In order to reduce copy-pasting, I want to create an environment that will produce a two-column table with headers and two listings in different languages, like below:  \begin{tabular}{l l} Python &amp; Haskell \\ \hline \\ \lstset{language=Python} \begin{lstlisting} A \end{lstlisting} &amp; \lstset{language=Haskell} \begin{lstlisting} B \end{lstlisting} \end{tabular}   I want to use it like below, hiding everything but code:  \begin{comparison} A \vs B \end{comparison}   So far I tried making two new environments (instead of a \vs command inside) with \lstnewenvironment, where first environment had \tabularx command and second \endtabularx. I also tried creating normal \newenvironment for this tabular, &amp; for \vs command and manually inserting \begin{lstlistings} and \end{lstlistings} with code, but it stopped compiling when I put lstlistings inside.  How can I create such an environment?  EDIT: The most compact version I've came up with until now is  \newenvironment{comparison} {\begin{parcolumns}[nofirstindent,distance=0pt,colwidths={1=0.35\textwidth}]{2}} {\end{parcolumns}}  \lstnewenvironment{python} {Python \lstset{language=Python,frame=t}} {}  \lstnewenvironment{haskell} {Haskell \lstset{language=Haskell,frame=t}} {}   which is used like that:  \begin{comparison} \colchunk{ \begin{python} A \end{python} } \colchunk{ \begin{haskell} B \end{haskell} } \end{comparison}   I'm still unable to hide \colchunk{} within python or haskell environments. Ubuntu 18.04 No WiFi adapter found [Dell Inspiron 3542.] What are the best SEO practices when changing domain names? I'm using the Yoast SEO plugin on a WordPress site on a new, more memorable domain to replace an old static HTML site with no SEO. I've followed , set up 301 redirects from the old domain to the new one, and filed a change of address in the Google Search Console. However, after a few days, the old site still shows in search results, and is ranked higher than the new one if you google our company name.   And, if I try to set the preferred domain in the Google Search Console Site Settings for the new site, I see this in place of the options to prefer www/non-www/http/https:     Not all options are available because of a Change of Address request   related to this site.   I've read of some cases where this condition has lasted for .  Is there anything more I can do to fix this? Have I done something wrong, or do I just need to wait, and for how long? A machine is described as being "really slow" or "freezing". How can I troubleshoot/resolve this issue? What should be the first steps to take? Find $x,y$ given $\gcd(x,y)$ and ${\rm lcm}(x,y)$ These two exercises I encountered recently seem to develop some type of connection between GCD and LCM I can't quite figure out. Exercise 1:  Find all the numbers $x$ and $y$ such that: $a) \ GCD(x,y)=15, \ LCM(x,y)=150$ $b) \ GCD(x,y)=120 \ LCM(x,y)=1320$ $c) \ GCD(x,y)=100 \ LCM(x,y)=990$  Exercise 2:   Find all the numbers $m,n$ such that $GCD(m,n)=pq , \ LCM(m,n)=p^2qs$  where $p,q,s$ are prime  The first thing that is known to me is that $GCD(x,y) \cdot LCM(x,y)= x \cdot y$ Also $LCM(x,y)$ is at most $x \cdot y$ while $GCD(x,y)$ is at most $\max \{x,y\}$. Last thing is that $GCD(x,y)|LCM(x,y)$. Using all this I tried to solve the first exercise: $a)$ First two obvious pairs are $x=15, y=150$ and $y=15, x=150$. Now neither of the numbers can be bigger than $150$ or smaller than $15$. So we are looking for numbers in the range $15-150$ that satisfy $x \cdot y = 15 \cdot 150$ Another such pair is $(x,y)=(30,75), \ (x,y)=(75,30)$. Similarly for $b)$ we find that the only possible values are permutations of the set {$120,1320$} and in $c)$ since $100$ does not divide $990$ no such numbers exist. Now exercise 2 is what made me think there is actually another connection I'm not quite aware of since now it's about arbitrary prime numbers and the previous method doesn't work anymore. My intuition is that it has something to do with $GCD$ or $LCM$ of the $GCD(x,y), \ LCM(x,y)$ Roman numerals used for ordinals How can one expand the function $f_1(x) = x$ on $(−π, π)$in terms of the functions $\cos nx, n = 0, 1, 2, ...$ and $\sin nx, n = 1, 2, ...,$ in a way so that $$f_2(x) = a_0 + \sum_{n=1}^{\infty}(a_n \cos nx + b_n \sin nx)$$ is the expansion of the $f_1$, how is that? I have asked  related to the same issue but didn't get full answer, I have been told that this is a standard application of Fourier series, $f_1$ and $f_2$ are equivalent in the sense that $f_1(x)=f_2(x)$ for all $x∈(−π,π)$. For any $x$ in this interval, the Fourier series will converge to $f(x)$, but I didn't get any rigorous proof  with elaborate explanation. So, plz provide rigorous proof of $f_1=f_2$ with elaborate explanation as I am new to the subject. Thanks You. I have attached the excerpt below - Let $R$ a ring, and let $$\displaystyle R[[x]]=\left\{\sum_{k=0}^{\infty}a_k x^k\;\middle\vert\; a_k\in R\right\}$$ with addition and multiplication as defined for polynomials.  We have that $R[[x]]$ is a ring containing $R[x]$ as a subring.  How to prove that if $a_0\in R$ is a unit, then  $\displaystyle\sum_{k=0}^{\infty}a_k x^k$ is a unit in $R[[x]]$? How to run a program without typing the full path? System utilities such as ls don't need to have the full path typed before executing them. However, to execute Netbeans, I have to type /home/chris/netbeans-7.0.1/bin/netbeans. How can I make it so that I only have to type netbeans? You can use the range feature of unicode-math to set the math font for a specific unicode slots or the macro accessing the glyph. I tried this with the sum sign &amp; Latin Modern, see my example:  \documentclass{scrartcl}  \usepackage{amsmath} \usepackage{unicode-math} % use unicode in math &amp; setup math fonts \setmathfont{Asana Math} %\setmathfont[range=\sum]{Latin Modern Math} % substitute Sum sign  \begin{document}  Notice the different sum sign $\sum$, as intended.   Notice also, the difference in spacing, not intended: \begin{equation}     \begin{bmatrix}         0           &amp; \frac{1}{2}   &amp; \frac{2}{3} \\         \frac{3}{4} &amp; 0             &amp; \frac{4}{5} \\         \frac{5}{6} &amp; \frac{6}{7}   &amp; 0     \end{bmatrix}     \begin{pmatrix}         A \\         B \\         C     \end{pmatrix} \end{equation} \end{document}   However, not only the sum sign, but also the spacing in the matrices changes. With Latin Modern commented out (as in the example):  With Latin Modern active (remove % before relevant line):  Why is that? BTW, I'm using Lualatex from TeXLive 2012. My fish's native habitat is rice fields. My fish's native habitat are rice fields.   Which one is correct? I'm pretty sure it's the first, since 'is' modifies 'habitat,' but it still sounds weird... a 2d planet in the center, where the players will be able to walk on it and also will be affected by center gravity. planet will be static and there won't be any other planets. so;  Is there a way that I can change the gravity position? yes you can adjust gravity via project settings but I want it to be in the middle x,y,z(0,0,0)  I have tried creating my own gravity but it wasn't very useful because objects wouldnt fall, stand on foot and also in contact with another object they would vibrate instead of standing still. (if they stand on the planet they would vibrate)          private void FixedUpdate() {     foreach(GameObject item in grav)     {         Attact(item);     } }  private void Attact(GameObject gmf) {     Rigidbody2D rgg = gmf.GetComponent&lt;Rigidbody2D&gt;();      Vector3 direction = rg2d.position - rgg.position;     float distance = direction.sqrMagnitude;      float forceMag = (10 * rgg.mass) / distance;     Vector3 force = direction.normalized * forceMag * 10;      rgg.AddForce(force); }   I can create a script that will change the gravity by the opposite of the object's position. this does work but I cannot see any way to do it for more than one object. In the original PC release of RollerCoaster Tycoon, if playing in a scenario with a lot of grassland, I would ensure my handymen did not mow the grass as their other services seemed more beneficial to keeping the park tidy, as well as making sure the handymen did not wander off the paths. In the android release, the handymen by default do not mow the grass (that duty by default is set to off).  I know the guests will remark on the scenery and vandalism, but I have not seen anything to suggest that mowing the grass has an effect on the guests or ride construction.  My question is; does mowing the grass have any effects apart from just making the park look tidier? Does allowing it to overgrow have any negative effects? Query to list number of records in each table in a database Shouldn't the badge requirements scale? Many badges have fixed requirements, such as:   Answer score of 100 or more   However different websites means a different level of difficulty: the more active users a website has, the easier the badge is to get.  On a new and/or low-view website getting this badge may be impossible for several months; on a website like stack overflow now you can obtain it in a day, and it's relatively easy (compared to other SE sites).  Do you think it would be a good idea to implement a formula that allows the required score to scale? The more active users voting, the more votes are required to get it?  If yes how would you handle people who got it for a post that no longer meets the requirements? Light Presets .exr I would like to install light presets for Blender to automate my lightning in my scenes. I have some .exr files (from Greyscalegorilla) but I don't know how to do it on Blender to make it work. Can someone help me? Thank you Why aren't bounties refunded if they are not awarded? I have  with a bounty that will expire in a few days. It looks like nobody will get the bounty; only one answer was posted during the bounty period, and it has no upvotes.  In cases like this, the bounty is thrown away.  Why isn't it refunded instead, if no answerer has earned it?