In case of two random variables $y,x$ we have that the best linear fit $y = \beta x + \epsilon$ satisfies  $$   \beta = \frac{\mathrm{Cov}(x,y)}{\mathrm{Cov}(x,x)}. $$ That is, if I known covariance matrix for $x$ and $y$ I can derive betas. Does the same argument apply also in case $x$ is multi dimensional?
After searching here:    I am still curious about this.  Is it possible to derive the standard error of a regression coefficient from summary data alone?   E.g., assume we are given the following variance-covariance matrix.  $\begin{bmatrix}Var(X) &amp; Cov(X,Y)\\Cov(X,Y) &amp; Var(Y)\end{bmatrix}$  We can derive the regression coefficient $\beta_{XY} = Cov(X,Y) / Var(X)$.  Given a specific n, is it possible to derive the standard error of $\beta$ as well? If so, which formula is being used? It appears that all formulas for regression standard errors that I could find assume that you know the variance of residuals of the regression, which we don't know from summary data alone.  ---UPDATE--- Based on feedback by Greg and whuber, I am now at a point, where I know that  $\sigma^2(b) = \sigma^2(X'X)^{-1}=\sigma^2(Var(X)(n-1))^{-1}$,  but typically matrix formulas for the standard error of $b$ then note that $\sigma^2$, the unknown variance of the errors, is estimated with MSE.   Based on comments by whuber, sum of squares residuals is:  $Y'Y - bX'X$, which according to Greg can be spelled out as  $Var(Y)(n-1) - b(Var(x)(n-1))$  And to go from here to MSE, I think (?) all that is left is divide by n-2,  so $(Var(Y)(n-1) - b(Var(x)(n-1))) / (n-2)$  And plugging this all in yields:  $\sigma^2(b) =((Var(Y)(n-1) - b(Var(X)(n-1))) / (n-2))\times(Var(X)(n-1))^{-1}$  I will try to see if this works out in R, and will report back. If anyone sees something blatantly wrong, I appreciate the heads up.
If it was possible to dig a hole that went from one side of the Earth to the other... ...And you jumped in.  What would happen when you got to the middle of the Earth? Would you gradually slow down, until you got to the middle and once you were in middle would every direction feel like it was up? I installed Ubuntu 12.10 and entered a password in SeaHorse. I would like to remove the password of "Default" but I get this error :  GDBus.Error:org.freedesktop.Secret.Error.NoSuchObject: The collection does not exist   Is there a fix or a way to bypass modify the password ? Which one is correct to say: "It's me" or "It's I"? The graduate students at my university (a relatively large state school in the US) are considering joining a union. To help decide whether to sign a union card in support of joining the union, I'm interested in more information about how unions have helped graduate students at other universities. What are some improvements to graduate life gained by graduate students at other universities upon joining a union?  I'm also interested in information about how unions have hurt graduate students at other universities. What are some detriments to graduate life caused by graduate students joining a union?  I'm primarily interested in student-body-wide benefits/detriments, instead of student-specific or faculty/administrator-specific claims (e.g. "my advisor treated me better with the union behind me" or "my students have stopped working since they joined a union" is not what I'm looking for). VLC will not install Can I use a HDD on my PS3 if I have already used another on the same console? I'm thinking of changing the Hard Disk of my PS3 (Slim 120GB), but I am unsure of what will happen to the data on the one I previously had if I do this:   PS3 has some data in the HDD. I turn off PS3, change HDD, install OS on new HDD. I download some data on new HDD from PS Plus.   Let's imagine I forgot to back something up...   I turn PS3 off, change HDD back.   Can I still use the old hard drive without needing to format it again? Alternatives to "and/or"? So I've been trying to create a mathematical model for an electric motorcycle and began to wonder about the maximum possible torque that could be supplied to the rear driven wheel without having the bike begin to lift up on one wheel.  I found ways to calculate this value online; however, the basic concept as to how the bike actually lifts escapes me.    My problem started when I began to think about what axis the bike will rotate about when it is doing the wheelie.  My first intuition was that the frame and front wheel, together, rotate about the rear axle.  But when I drew a free body diagram of the frame/front wheel system just at liftoff (see below) I made note that the only forces acting about the rear axle, point O, is the force of weight.  This means that an increase in applied torque and subsequently, the applied force Fa, should not effect the rotation about the rear axle.  I know that the applied force on the back wheel is indeed correlated with the propensity for a bike to wheelie, so I considered that the axis of rotation I was choosing was wrong.  If we take the free body diagram above and sum the moments about the center of mass, we would find that an increased applied force would in fact cause the solid body to rotate.  The problem with this understanding is that during rotation, the center of mass of the drawn system should actually rises relative to the surface the bike is moving across.  If the bike were to be truly rotating about its center of mass, then the back wheel would begin to dip below the surface of the road like you might see in a glitchy video game.    So I suspect that the bike is in fact rotating about the back axle, but I don't understand why, please help!  edit: I added the external torque from the back wheel to the frame, which would allow the bike to rotate about the back axle  edit 2:  I suppose that the torque acting on the frame via the engine should not effect the rotation as the movement of a motorcycle can be perfectly replicated by applying a force at the back axle.  A third possible axis of rotation might be the lowest point of the back wheel. Evaluation of $\lim_{n\rightarrow \infty}\frac{n!\cdot e^n}{\sqrt{n}n^n}$ without stirling approximation Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\frac{n!\cdot e^n}{\sqrt{n}\cdot n^n}$   $\bf{My\; Try::}$we can write it as $$l=\lim_{n\rightarrow \infty}\frac{e^n}{\sqrt{n}}\cdot \left(\frac{1}{n}\cdot \frac{2}{n}\cdot \frac{3}{n}\cdots \cdots \frac{n}{n}\right)$$  $$\ln (l) = \lim_{n\rightarrow \infty}\bigg[n-\frac{1}{2}n+\sum^{n}_{r=1}\ln\left(\frac{r}{n}\right)\bigg]$$  Now how can i solve it, Help required, Thanks Do you use NULL or 0 (zero) for pointers in C++? I'm looking for an algorithm which when given a latitude and longitude pair and a vector translation in meters in Cartesian coordinates (x,y) would give me a new coordinate. Sort of like a reverse Haversine. I could also work with a distance and a heading transformation, but this would probably be slower and not as accurate. Ideally, the algorithm should be fast as I'm working on an embedded system. Accuracy is not critical, within 10 meters would be good. What does each AMS package do? In my view, any document with a substantial amount of mathematics should load the packages from AMS-TeX. Hence, most of my document's preambles contain  \usepackage{amsmath, amsthm, amssymb, amsfonts}   (I could add to that the non-AMS mathtools, etc.). What do each of those packages do? Do I really need to list all of them, or do some of them load others? In particular, does any package load all of them, so that I can simply type  \usepackage{ams-everything}   shorter than the current list? One of my advisors suddenly passed away while I was in graduate school.  We had some discussions and ideas about future publications, but he passed away before any of the work was completed.  When the work was finally completed and published, I and my co-authors were therefore presented with an ethical dilemma about how best to acknowledge his contributions to the ideas behind the paper.  Should we list him as a co-author?  Put him in the acknowledgements?  Listing him as an author would give credit for the original idea, however, we would have no way of knowing if he actually approved of—and would want his name attached to—our methods and writing.  In the end my co-authors and I decided to list him as a co-author with a footnote stating that he passed away before publication.  I’m interested to hear from others who have been in similar situations and/or suggestions on what constitutes “co-authorship” when one of one’s collaborators passes away before the publication or work is complete. Proof of the square root inequality $2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}$ A 398MB directory was only compressed to 393MB using 7Z and Normal ZIP compression. Is this normal? If so, why do people continue to use ZIP on Windows? 'Possible duplicate' comment not appearing for questions with deleted comments containing the link How can I play around targeted discard? Many decks in competitive Magic have access to targeted discard spells, like  and .  Against counterspells, you can rely on a variety of tricks like baiting counters, bottlenecking your opponent's mana, and good old-fashioned counter wars. Targeted discard cards allow your opponent to attack your hand on the very first turn of the game, and go right for your key cards &mdash; and it can often be backbreaking when your opponent takes away a crucial card like a combo piece or a sweeper.  What tricks and tactics can you employ, either during deck-building or in play, to play around an opponent's targeted discard?  (I know that one of the answers is going to be . That's excellent, but please also include approaches that work in formats without free countermagic.) I was reading a book and this sentence seemed a little odd to me:     At approximately 19.45 hours the two men rose, stretched and yawned. They picked up their gear and stood at the door, ...   What does the first part mean? If it's the exact time, shouldn't it be 19:45? And if it's duration, shouldn't it be "After 19.45 hours"?  (The story here happens at an airbase, if it's of any relevance)    Edit: The book I've quoted is this:   by John Creed.   The author is Irish and the text is set in northern England/Ireland. Showing that $e^x > 1+x$ Maximum sum sublist?