For $x_1,..,x_n \in \mathbb{R}$, what is the value of $\mu$ that minimizes $\sum_{i = 1}^{n}|x_i-\mu|$? What about $\sum_{i = 1}^{n} w_i |x_i-\mu|$ for $w_i \geq 0$?  I'm pretty sure in both cases the answer is the average of the $x_i$, and when you introduce the weights it's just the weighted average.   I'm not sure how to prove this though, and would appreciate any help.
Suppose we have a set $S$ of real numbers. Show that   $$\sum_{s\in S}|s-x| $$   is minimal if $x$ is equal to the median.    This is a sample exam question of one of the exams that I need to take and I don't know how to proceed.
The limit of a convergent Gaussian random variable sequence is still a Gaussian random variable I'm trying to prove this conclusion but have some problems with one of the steps.  Assume $X_1,\ldots,X_n,\ldots$ is a sequence of Gaussian random variables, converging almost surely to $X$, prove that $X$ is Gaussian.  We use characteristics function here. Since $|\phi_{X_n}(t)|\leq 1$, by dominated convergent theorem, we have for any $t$  $$ \lim_{n\rightarrow\infty}e^{it\mu_n-t^2\sigma_n^2/2}=\lim_{n\rightarrow \infty}\phi_{X_n}(t) = \lim_{n\rightarrow \infty}\mathbb{E}\left[e^{itX_n}\right] = \mathbb{E}\left[e^{itX}\right] = \phi_X(t) $$  this is the step that I cannot figure out: $e^{it\mu_n-t^2\sigma_n^2/2}$ converges for any $t$ if and only if $\mu_n$ and $\sigma_n$ converges.   Let $\mu=\lim_n \mu_n$, and $\sigma=\lim_n\sigma_n$, then $\phi_X(t)=e^{it\mu-t^2\sigma^2/2}$, which proves that $X$ is a Gaussian random variable.  Why can we get that $\mu_n$ and $\sigma_n$ converge? This looks intuitive for me, but I cannot make a rigorous prove. Is there anyway to find and remove ligatures from tex file? Sometimes when I copy/paste a text to Winedt, it shows perfect, but when I compile it to Pdf, some issues appear, e.g. finding becomes nding.  I am wondering if there is any way to search and find these characters?  Here is an example file (needs utf8 encoding).  \documentclass{article} \begin{document} ﬁ \end{document} We all know very well how to make an angle of 15 and 45 degree and its multiple using compass. Can anybody tell me how to make an angle of 10 degree and its multiple using compass? I am using Google Spreadsheets to create a matrix (time/activity) as a daily schedule for staff to sign into.  Is there a formula to calculate the number of cells in a column that are shaded (or not shaded) a certain color?  For example: 20 of the cells are gray and 31 are not gray (shaded any other color) and instead of manually counting them, it calculates this at the bottom of reach column. How to make beveled 90 degree corners on a curve better? I know the title is a mouthful, but this is the problem: I made a curve with 90 degree corners, like so:  I then beveled it, and this was the result:  As you can imagine, this isn't very pretty. I'm looking for something like this:  How can I achieve this look? Is it possible to configure Windows 7 to use YYYY-MM-DD instead of DD/MM/YYYY as the date format? How? Where? Is there any physical or mental cost to casting spells? What regression model is the most appropriate to use with count data? I am trying to get a little into statistics, but I am stuck with something.  My data are as follows:  Year   Number_of_genes 1990          1 1991          1 1993          3 1995          4   I now want to build a regression model to be able to predict the number of genes for any given year based on the data. I did it with linear regression until now, but I have done some reading and it does not seem to be the best choice for this kind of data. I have read that Poisson regression might be useful, but I am unsure what to use. So my question is:  Is there a general regression model for this kind of data? If no, what do I have to do to find out which method is the most  appropriate to use ( in terms of what I have to find out about the data)? I sometimes have long running processes that I want to kick off before going home, so I create a SSH session to the server to start the process, but then I want to close my laptop and go home and later, after dinner, I want to check on the process that I started before leaving work. How can I do that with SSH? My understanding is that if you break your SSH connection you will also break your login session on the server, therefore killing the long running process. Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$. How to change Ubuntu 18.04 system fonts? I installed Ubuntu 18.04, and I want to customize the fonts. I have already change all the setting by tweak tool. But the system default font not be changed, is anything I need to do? Separate selected UV map vertices from surrounding vertices How do I break the selected UV map from its surrounding vertices? What are the SEO rules for HTML 5? What are the rules of engagement with respect to SEO in HTML 5?  Examples of things we have been mulling over with no conclusion:   Are you penalised for not using &lt;head&gt; and &lt;body&gt; tags? Do links in &lt;nav&gt; blocks get prioritised for follows? With &lt;aside&gt; (no bside though), &lt;section&gt;, &lt;summary&gt; and &lt;details&gt; which ones carry the most weight for content, one-liner descriptions and keyword tags are there search engine implications? If you use CSS to put a &lt;footer&gt; element at the top and a &lt;header&gt; element at the bottom (not that you would but it may be construed as such by an incomplete spider implementation)? Does the use of placeholder attributes on &lt;input&gt; show you are more user focused?   At the minute we are haphazardly pinning the content on the donkey with regard to tag usage but would be interesting to know if anyone has any insight into the "correct" tags to use with regards to SEO. I am using gdal_rasterize to create a raster of a huge file containing polygons. I managed to generate the raster file with the right color and with the minimum size to tile it after with the gdal2tile.  gdal_rasterize -of GTiff -co COMPRESS=DEFLATE  -b 1 -b 2 -b 3 -burn 136-burn 46 -burn 17 -co PREDICTOR=2  -co ZLEVEL=9 -ot Byte  -a_nodata 255  -a Niveau  -ts 500000 500000  -l layer W:/layer.shp W:/layer.tif   Tile will be used to overlay Google Maps so the polygon must be transparent.  Has anybody experimented with this usage of gdal_rasterize and succeeded in setting the opacity of the output raster? Capitalize fields of study? I don't know anything about Star Trek except for the fact that it's about spaceships.  In  the person only wanted to watch the TV series. I want to watch every movie and TV series there is about Star Trek but I don't know where to start. I found this page on  which is about a time line. Do I need to watch it in this order? Are the movies and TV series related? Can Wolverine survive without oxygen? . Even poisons don't work on him. He seems immortal due to his regeneration powers.  Can he survive without oxygen while his body exists in full working state (I am not talking about skeleton-only Wolverine when his respiratory system is not available)? Too many of your edits were rejected, try again in 7 days Defining the Product of Ideals Where does the &quot;naive definition&quot; of the product of two ideals $I$ and $J$, $IJ = \{ ij \mid i \in I, j \in J \}$ fall apart? (The product of two ideals $I$ and $J$ is defined to be $IJ := \sum_i a_ib_i$, where each $a_i$ is in $I$ and each $b_i$ is in $J$.)  Note: This is a question from a problem set I'm working on. I am not expecting a full solution for the answers, but I'd like some help knowing where to look. It seems to work fine in $\mathbb{Z}$ and $\mathbb{C}[x]$. How to shift EXIF date/time created by time in days, hours, minutes?