I was reading from The Elements of Statistical Learning (Section 3.2) and they were able to find $Var(\hat\beta)$ Given $$\hat\beta = (X^TX)^{-1}X^Ty \qquad (3.6) $$ The answer they had was $Var(\hat\beta) = (X^TX)^{-1}\sigma^2$ Please how do I find $Var(\hat\beta)$ from $ (3.6)$
I just read this very insightful post , where the author stated that the variance of $\hat\beta$ is:  $$\text{var}(\hat\beta) = \sigma^2(\textbf{X}^\prime \textbf{X})^{-1}.$$  I couldn't figure out why it is like this. Can anyone elaborate a bit?
What is the life expectancy of a Wookiee? At lunch, my coworkers and I were discussing The Force Awakens, and started questioning Chewbacca's age.  A quick check of  and it says he is possibly about 230 years old as of Episode VII.  Following the  link on that page, Wookieepedia claims that they live to be 400 years old.  Do we know from canon what the average life expectancy of a Wookiee is? $\textit{Why}$ the cases in 4 $\nmid n^2 - 3?$ (bases cases in modular induction) This question has already been answered . My question is, when we suppose the hypothesis is not true (proof by contradiction), namely that $4 \mid (n^2 - 3)$, we get $n^2 - 3 = 4k$ for some integer $k$ and then we have two cases. But where does the two cases come from? In addition, why does putting them into two cases work to prove the statment? My thought process would be to try to prove this statement by contradiction and somehow figure out a way to produce a contradiction similar to how $\sqrt{2}$ is proved by contradiction. Any help is great! Let me know if I need to clarify more! What EXACTLY happens when I ask to delete my account? Explanation of finite correction factor Is the product of two non-holomorphic function always non-holomorphic? Suppose $h(z)$ is a complex function. I have noticed that $h(z) = f(z)\cdot g(z)$, where $f(z)$ and $g(z)$ are non-holomorphic. Can $h(z)$ be holomorphic?   Can a similar statement be made, if $f(z)$ is holomorphic, but $g(z)$ is not? What happens to matter in a standard model with zero Higgs VEV? Sending a value to JavaScript I have set theme handler for a form in Drupal and this handler has a template file. What is the easiest way to send a value from PHP to JavaScript in that include file? Difference between standard beta and unstandard beta distributions? How to label using just numbers from start of text field? In ArcMAP 10.2, I am trying to format the labels on my map.   If the label is   1402 Bowie Road  But  I want only to display numbers:  1402   What is the expression to create this? Add a binary to my path I have an executable. I want to execute the executable in terminal with name only like other commands.  I can put my executable in /usr/local/bin or I could add its PATH to ~/.bashrc. Both will work.  What is better?  Is there  any difference? No Wifi-Adapter found on new Lenovo Yoga C930 with Ubuntu 18.10 How to properly set up a PDO connection How can I prove that if $G$ is an Abelian group with elements $a$ and $b$ with orders $m$ and $n$, respectively, then $G$ contains an element whose order is the least common multiple of $m$ and $n$?  It's an exercise from Hungerford's book, but it's not homework. I could not solve it when I take the course on groups and I think it should be easy. There is a hint that says to first prove when $m$ and $n$ are coprimes. I did this part. But I have no idea how to solve the general case.  Thanks. The banner ads are in the wrong format for the right-hand ad slot:    These kinds of ads are not useful (because most of it is cropped) and should be excluded from being shown in that spot. I understand that dark matter does not collapse into dense objects like stars apparently because it is non-interacting or radiating and thus cannot lose energy as it collapses. However why then does it form galactic halos? Isn't that also an example of gravitational collapse? The Cartesian product of a family $(A_i\mid i\in I)$ is defined as $$\prod\limits_{i\in I}A_i=\{f:I\to\bigcup A_i\mid f(i)\in A_i\}$$ Let $I_n=\{i \in \mathbb N \mid i \le n\}$ for $n\in \Bbb N$ and $A_i$ be countable for all $i\in I_n$. Then $\prod\limits_{i\in I_n}A_i$ is countable.   My attempt: Lemma 1: $(\prod\limits_{i\in I_n}A_i) \times A_{n+1}$ and $\prod\limits_{i\in I_{n+1}}A_i$ are equinumerous for all $n \in \mathbb N$. (I presented a proof ) Lemma 2: If $A$ and $B$ are countable, then $A \times B$ is countable. (I presented a proof ) I will prove the theorem by induction on $n$. The theorem is trivially true for $n=0$. Assume that the theorem is true for $n=k$, then $\prod\limits_{i\in I_k}A_i$ is countable. Since $\prod\limits_{i\in I_k}A_i$ is countable (by inductive hypothesis) and $A_{k+1}$ is countable, then by Lemma 2 $(\prod\limits_{i\in I_n}A_i) \times A_{n+1}$ is countable. Furthermore, by Lemma 1 $(\prod\limits_{i\in I_k}A_i) \times A_{k+1}$ and $\prod\limits_{i\in I_{k+1}}A_i$ are equinumerous, then $\prod\limits_{i\in I_{k+1}}A_i$ is countable too. Thus the theorem is true for $n=k+1$. This completes the proof.   Does this proof look fine or contain gaps? Do you have suggestions? Many thanks for your dedicated help! Accessing UI thread handler from a service Baryons have charges that are the result of a polynomial calculation of their building blocks (quarks)'s fractional charges. But what gives these quarks electric charges? What interactions do they have with photons? And what about leptons, like muons and electrons? How do they get their -1 charge? And then, how do positrons and anti-muons get +1 charge? Same goes for quarks: how does up-antiquarks get -2/3 charge? Or I may want to formulate my question this way:  How does any elementary particle interact with the EM-Force to get an electric charge? Maven plugin executing another plugin I'm trying to create a new plugin to package my latest project.  I'm trying to have this plugin depend on the maven-dependency-plugin to copy all my projects dependencies.  I've added this dependency to my plugin's pom, but I can't get it to execute.  I have this annotation in my plugins main Mojo:  @execute goal="org.apache.maven.plugins:maven-dependency-plugin:copy"   I've tried a few other names for the goal, like dependency:copy and just copy but they all end with a message saying that the required goal was not found in my plugin.  What am I doing wrong?  Secondary to this is where to I provide configuration info for the dependency plugin? Installing TTF fonts in LaTeX One of the annoying aspects of LaTeX is the limited number of fonts that come by default, and the pain involved in making new fonts 'LaTeX' ready. I have a collection of truetype fonts that I'd like to prepare for use, and I definitely want to make sure I have vector versions of these fonts (i.e not type 3/bitmapped versions). Is there a relatively painless way to do this ?