What does $(X'X)^{-1}$ matrix mean in regression? In a regression what is the matrix $(X'X)^{-1}$? What do its interior elements represent?
Are estimates of regression coefficients uncorrelated? Consider a simple regression (normality not assumed): $$Y_i = a + b X_i + e_i,$$ where $e_i$ is with mean $0$ and standard deviation $\sigma$. Are the Least Square Estimates of $a$ and $b$ uncorrelated?
Adverb sentence query Is there an IDictionary implementation that, on missing key, returns the default value instead of throwing? The indexer into Dictionary throws an exception if the key is missing.  Is there an implementation of IDictionary that instead will return default(T)? I know about the TryGetValue() method, but that's impossible to use with LINQ. Would this efficiently do what I need?: myDict.FirstOrDefault(a =&gt; a.Key == someKeyKalue);  I don't think it will as I think it will iterate the keys instead of using a Hash lookup. What is a good, short, word to describe a software engineer? What should I call a person who write software, computer programs ? I know he/she is a software engineer, can I call him/her as "Coder"? Get image pixels of prescribed color I am flying from Orange County, California to San Francisco, California and then on to Seoul, South Korea. If I check a bag will I have to pick it up in San Francisco and re-check it and go through security or customs for the international flight?  Everything I am reading leads me to believe that it's only necessary to re-check baggage and go through security and/or customs when you are first arriving into a country. However, since my first leg is staying within the United States I assume the airline will transfer my bag to the next flight and I can just go straight to the gate for my international flight?  Both legs of the flight are through the same airline (United Airlines) and there are only the two legs: OC -> SF, then SF -> Seoul. Is it okay to use my US visa after a name change? I had my US visa about 7 month age, and I had to renew my passport for certain circumstances, so there was a mismatch in the name between both the visa and the new passport, you see they had to put my grand grandfather's name which was not on the old passport nor the visa, the name on old passport and visa is: Abdelrahman Samy Ali Mohammed and the new passport is: Abdelrahman Samy Ali Ibrahim Mohammed. My grand grandfather's name is Ibrahim, so the question is that gonna cause me any troubles concerning my entry to the US? is there any document should I consider taking it while i'm traveling?  Thanks,, I know that if the moment generating function of two distribution converges to the same function then the two distribution converges in CDF. But how can we prove this thing explicitly ? CSS background opacity with rgba not working in IE 8 On the nature of the collapse of the wave function The collapse of the wave function by measurements is one of the most mysterious properties of quantum mechanics.    At what scale does the wave function collapse? What are the conditions for a collapse?  Can it be made precise?  Does the collapse happen instantaneously, or is it spread out in time?  What is the precise equation for the collapse?  Why is the collapse time-asymmetric, and can this explain the arrow of time? How to resolve "iTunes could not back up the iPhone because the backup was corrupt"? When I try to sync my iPhone with my computer, I get the following message:       iTunes could not back up the iPhone "iPhone Name"   because the backup was corrupt or not compatible   with the iPhone.      Delete the backup for this iPhone, then try again.   Yet, here is no option to delete the backup. Where are backups stored / how can I delete the backup?  This is iTunes version 10.5.3 running under  Mac OS X 10.6.8. I have already seen a proof of this statement here on stackexchange but I would like someone to verify whether "my proof" is correct and also if you could point out issues with the writing style. thanks in advance!  Statement. $\mathbb{Z}_{p}$ is an integral domain $\Leftrightarrow$ $p$ is prime.  Proof.$(\Rightarrow)$ Suppose $\mathbb{Z}_{p}$ is an integral domain. Assume (by way of contradiction) that $n$ is not prime. Then $n=ab$ for some $a,b\in \mathbb{Z}$ where $1&lt;a$ and $b&lt;n$, meaning $ab\equiv 0\pmod n$. This is a contradiction since $\mathbb{Z}_{p}$ is assumed to be an integral domain and therefore does not contain any zero divisors. Therefore $p$ must be prime.  $(\Leftarrow)$ Assume $p$ is prime. Then $a$ does not divide $n$ for any $1&lt;a&lt;n$ where $a\in \mathbb{Z}_{p}$. Thus $\mathbb{Z}_{p}$ does not contain any zero divisors and is therefore an integral domain. Why aren't these loops homotopic? Let $S^1 = \{z \in \mathbb{C} : |z| = 1\}$.  Take the loops $f,g : [0,1] \rightarrow S^1$, $f(t) = 1$, $g(t) = e^{2\pi it}$.  I know these represent different elements in $\pi_1(S^1, 1)$, but I don't see why $F(t,s) = e^{2\pi its}$ isn't a homotopy between $f$ and $g$. In Java, how do I call a base class's method from the overriding method in a derived class? While trying some problems along with my friends we had difficulty in this question.   True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ is $1$.   I couldn't do it and my friend justified it by saying that since the terms in the product have values less than $1$, so the value of the product can never be $1$. I don't know whether this justification is correct or not. But i referred to Tom Apostol's Mathematical Analysis book and found a theorem which states, that    The infinite product $\prod(1-a_{n})$ converges if the series $\sum a_{n}$ converges.   This assures that the above product converges. Could anyone help me in finding out where it converges to? And,   Does there exist a function $f$ in $\mathbb{N}$ ( like $n^{2}$, $n^{3}$) such that $\displaystyle \prod\limits_{n=1}^{\infty} \Bigl(1-\frac{1}{f(n)}\Bigr)$ has the value $1$? If I have a sequence of .obj (or any other readable format) files created from my own simulation code (say cloth simulation), can I use the rendering capabilities of blender to generate a video?   Is there a way to set the material properties for frame 1 and use the same or rest of the frames while the corresponding meshes are imported afresh from that ones I provide? Alternative definition of hyperbolic cosine without relying on exponential function Ordinary trigonometric functions are defined independently of exponential function, and then shown to be related to it by Euler's formula.   Can one define hyperbolic cosine so that the formula $$\cosh{x}=\dfrac{e^x+e^{-x}}{2}$$ becomes something to be proven? We (Russian citizens) are planning to go to Munich in late June. We would like to book our accommodation on AirBnB, but it is not very clear whether it will be accepted by the German embassy when we apply for a Schengen visa.  I can't find such information on the German embassy site or on the VFS-Global site. I've also searched for it on the Internet, and, while most sites agree that AirBnB booking is generally OK for a visa, I don't see a clear answer for Germany specifically, and there seems to be some exceptions for other European countries. Chrome phishing warning on main The derivation of the Hamilton-Jacobi equation using canonical transformations is typically done involving a type-2 generating function.  Is it possible to use a another type of generating function, namely type 1, 3 or 4? If light is an electric and (magnetic field), how can it be absorbed? I was wondering how light or any electromagnetic radiation can be "absorbed" if it consists of electric and magnetic fields. For example if there is a charge at point A, and the  light reaches point A, it will cause that charge to move depending on the value of the electric field. However, why wouldn't this electric field continue on and influence others particles everywhere else? For example, if you have a static charge creating an electric field, then if you put a charge at point A it will feel its effects and then if you bring a charge at another location B it will also feel its effects, namely it will not have been absorbed. Please shed some light on this. Thanks  edit: Does this mean the electron that absorbs this light moves in such a way that it creates an electric field of its own canceling the initial one? Or am I way off?