Ridge Regression: When should the intercept be included ? What is the purpose of the intercept term?
When is it ok to remove the intercept in a linear regression model?
I set up a new VPS instance of Ubuntu and am wondering what the easiest way is to get up and running with a basic LAMP stack (i.e. which packages are required, which configuration options need to be tweaked, if any, etc.). If $X$ is an orthogonal matrix, why does $X^TX = I$? It's not immediately clear to me why this is true. My notes say that putting $n$ orthonormal vectors $ v_1, ..., v_n$ in the columns of $X$ gives $X^TX = I$, and it follows from this that the rows of $X$ are orthonormal.   Can you explain this better to me? I tried the $2\times 2$ case, but it isn't clear to me why this is true. We were asked to prove the following:  Let $ A $ be an $n \times n$ matrix with $n$ distinct real eigenvalues. If $AB=BA$, show that $B$ is diagonalizable.   It was suggested I show that an eigenvector of $A$ is also an eigenvector of $B$. I am both having trouble doing this and failing to see how I would complete the proof after. Any help would be appreciated.  Thanks How do you keep parents of floated elements from collapsing? Make the MSE favicon grayscale (like all other meta icons are) The new  for Meta Stack Exchange looks exactly the same as the one for the  site:        This can get a bit confusing, especially if you happen to have many tabs open from both sites.  It also looks a bit silly in the site switcher menu:  All the per-site metas (and MSO, even before the MSO/MSE split) have favicons that are grayscale versions of the main site icons.  Why not do the same for MSE? Conveniently, michaelb958 already wrote a nice  to do just that (and I  it to apply to the menu too), so you can easily see for yourself how it would look.  Or just see the screenshot below:  So here's my proposal: Let's make this gray icon the official MSE favicon.  Update: The meta.SE favicon has recently been changed to a simplified outline version of the SE logo:  I'm not sure how much I like the new logo (for one thing, it doesn't seem to match the dimensions of the main SE logo), but at least it's different, which was the main point of this request. I have seen people showing one slide progressively by pressing Enter. Could anyone tell me how to realize that in Beamer? I called my lender last week. I am close to owning 20% of my home and want to get rid of PMI. They said I can only do it by remortgaging my house. Is that true? With the mortgage rates increasing, I will definitely lose some money by remortgaging the house.  Also, I know that I can request the PMI charge to stop if I own 20% of the value of the home, or it will stop automatically once I own 22% of the value. Is it possible that the lender meant I can't do it with 20% but it will be done at 22% automatically without the need to remortgage the house? Axis with trigonometric labels in PGFPlots How do I make the axis labels use multiples of \pi in PGFPlots? Dual-boot boot menu does not show up after installing Ubuntu 15.10 alongside Windows 10 I have installed Ubuntu 15.10 alongside Windows 10 with UEFI. To install Ubuntu, I chose the option install alongside Windows 10 or something similar to this. Then I created a new partition for Ubuntu and installed it. After installation, the boot menu did not show up. Initially I thought Ubuntu has not been installed, but when I plugged in the USB drive and wanted to install Ubuntu I saw an option of reinstalling Ubuntu on my machine. So, I found out that Ubuntu is installed. Pressing F8 and F12 also does not help. Can anyone help me bring up the GRUB boot menu? In Windows, I also entered the command bcdedit /set {bootmgr} path \EFI\ubuntu\grubx64.efi in cmd, but still the boot menu does not show up.  Reply (Himanshu) : Just boot in bios and, if you can, add boot option with name say ubuntu and path EFI/ubuntu/shimx64.efi. No need for live PCB or anything. Move the boot option to top. This is for dell GUI bios but I assume it works for all. Or you may want to see  or , where you can enable Windows Boot Loader for one boot or forever as you wish, and boot Ubuntu from it. If you want now, you can use the installed Ubuntu terminal to use the commands update-grub to use GRUB instead. (Not enough reputation to answer properly btw.) Using jQuery, how do I change the elements html value? (div) Using a ajax request I want to change content of my div.  &lt;div id="d1"&gt;202&lt;/div&gt;   So I want to change the content to a different number.  $('d1').InnerText???   Also, say I wanted to increment the number, how could I do that? Do I have to convert to int? Pros and cons of AppSettings vs applicationSettings (.NET app.config / Web.config) When developing a .NET Windows Forms Application we have the choice between those App.config tags to store our configuration values. Which one is better?  &lt;configuration&gt;    &lt;!-- Choice 1 --&gt;   &lt;appSettings&gt;     &lt;add key="RequestTimeoutInMilliseconds" value="10000"/&gt;   &lt;/appSettings&gt;    &lt;!-- Choice 2 --&gt;   &lt;configSections&gt;     &lt;sectionGroup name="applicationSettings" type="System.Configuration.ApplicationSettingsGroup, System, Version=2.0.0.0, Culture=neutral, PublicKeyToken=b77a5c5612342342" &gt;         &lt;section name="Project1.Properties.Settings" type="System.Configuration.ClientSettingsSection, System, Version=2.0.0.0, Culture=neutral, PublicKeyToken=b77a5c5612342342" requirePermission="false" /&gt;     &lt;/sectionGroup&gt;   &lt;/configSections&gt;   &lt;applicationSettings&gt;     &lt;Project1.Properties.Settings&gt;       &lt;setting name="TABLEA" serializeAs="String"&gt;         &lt;value&gt;TABLEA&lt;/value&gt;       &lt;/setting&gt;     &lt;/Project1.Properties.Settings&gt;   &lt;/applicationSettings&gt;  &lt;/configuration&gt; Can I adjust starting point when using ListPlot? Indian passport threading coming off My Indian passport threading is coming off i.e. the cover of the passport is coming off, there are only 2 threads remaining.   Will it be a problem at the airport when going abroad ? Also anyone who has experience of getting a replacement passport fairly quick ? Because I need to get a passport in like 15 days. What happened to the “‑est” and “‑eth” verb suffixes in English? What happened to them, and how were they once used? Straining my mind to sound archaic, I came up with the following:     Dost thou thinkest thou can escape thy sins?   and     Bringeth me mine armor and favorite sword.   I’d like to use these suffixes intelligently, so my questions are: how are ‑est and ‑eth properly appellated in conjugations, and  when and why did they disappear? Reference — frequently asked questions about PDO How do I calculate the probability density function for a joint beta + uniform distribution? The PDF of a Beta distribution is $$ f_X(x) = {{x^{a-1}(1-x)^{b-1}}\over {B(a,b)}}  $$ and CDF $$ F_X(x) =  I_x(a,b) $$ The PDF of a uniform distribution is $$ f_Y(y) =  {1 \over {b-a}} \space for \space  x \in [a,b] $$ and CDF $$ { x - a} \over {b - a} $$ I've know that if X and Y are two independent, continuous random variables, described by probability density functions above of $f_X(x)$ and $f_Y(x)$ then probability density function of $Z = XY$ is $$  f_Z(z) = \int_\infty^\infty f_X(x)f_Y({z\over x}){1\over abs(x)}dx  $$ but I'm struggling with the algebra to get  from this to the PDF and the CDF of the joint distribution. I'm using the uniform distribution between 0 and 1. How exactly was the finalist chosen in the NIST AES competition? Is every compact subset of $\Bbb{R}$ the support of some Borel measure? I have tried to prove the exercise 2.12 in Rudin's RCA:     12 Show that every compact subset of $\Bbb{R}$ is the support of a Borel measure.   For perfect (i.e. no isolated point) compact $K$ with non-zero Lebesgue measure, we just take the Lebesgue measure. However I struck on the general case.  My attempt is as follows: For every compact $K$, we can find a decreasing sequence of open set $\langle W_n\rangle_n$ s.t. $\bigcap_n W_n =K$ and $m(W_n-K)&lt;1/n$. Find a continuous function $h_n$ s.t.   $h_n(x) = 1$ for $x\in K$ and $h_n(x) = 0$ outside of $V_n$.   (such function exists by Urysohn's lemma.) For continuous function $f$ with compact support define $$\Lambda f = \lim_{n\to\infty}\frac{\int_\Bbb{R} fh_ndm}{\int_\Bbb{R} h_ndm}.$$  I can not certain that the $\Lambda$ is well-defined. But if it is well-defined, then it gives an regular Borel measure (by Riesz representation theorem) $\mu$ s.t. $\mu(K)=1$. I guess that the $\mu$ has a support $K$ but I don't know how to get it. Thanks for any help! How can Linux/Ubuntu be installed safely on ChromeBook? Is installing Ubuntu on my Chromebook potentially dangerous to anything on my Chromebook? I got it as a gift, so I don't want to destroy anything that is important on the laptop. Also, which is better for being able to safely install Ubuntu: ChrUbuntu or Crouton? I'm using the Samsung Chromebook 3 (as described by the quick-start manual). Does that make any other differences as well? Determine all of the accumulation points of the following sets in $\mathbb{R}^1$ and decide whether the sets are open or closed or neither.  The set $X$ of all numbers of the form $2^{-n} + 5^{-m}$ where $m,n = 1,2,\ldots$.   Claim: The set of accumulation points is the following: $$\left\{\frac{1}{2^n}: n \in \mathbb{N} \right\} \cup \left\{\frac{1}{5^n}: n \in \mathbb{N}\right\} \cup \{0\}.$$ Let $n \in \Bbb{N}$ and $r&gt;0$, and consider the open neighbourhood $B\left(\frac{1}{2^n},r\right) = \left(\frac{1}{2^n} - r,\frac{1}{2^n} + r\right)$. For the given $r$ we can find $m$ large enough such that $5^{-m} &lt; r \implies \frac{1}{2^n} + \frac{1}{5^m} &lt; \frac{1}{2^n} + r$, and $\frac{1}{2^n} - \frac{1}{5^m} &gt; \frac{1}{2^n} - r$; therefore $5^{-m} + 2^{-n} \in B\left(\frac{1}{2^n},r\right)$. A similar argument shows that each element of $\{5^{-m}: m \in \mathbb{N}\} \cup \{0\}$ is an accumulation point of $X$. Now consider $y \in X$. $B(y,r)$ isn't a subset of $X$ since for all $r &gt; 0$, $B(y,r) \subset (\mathbb{R} - \mathbb{Q}) \not\subset \Bbb{Q}$, hence $B(y,r) \not\subset X$.