Do characteristic functions characterize the independence of random variables? [Solved] It is well known that the probability density function characterizes the independence of random variables in the following sense.  $$X,Y \quad\text{independent}\iff f(x,y)=f_x(x)f_y(y)$$  where $f$ is the pdf of the random variable $(X,Y)$ and $f_x$ and $f_y$ are the pdfs of the marginal distributions of $X$ and $Y$, respectively. My question is, seeing as there is a bijection between characteristic functions and probability distributions, do characteristic functions also characterize independence? In the following sense  $$\varphi_{(X,Y)}(t_1,t_2)=\varphi_X(t_1)\varphi_Y(t_2)$$  where again $\varphi$ is the characteristic function of the random variable in its subscript.
Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent. This is solely a reference request. I have heard a few versions of the following theorem:     If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ whenever the expectations are finite, then $X,Y$ are independent.   And there is a similar version for characteristic functions. Could anyone provide me a serious reference which proves one or both of these theorems?
I have a normal door, and left to its own devices it will simply rest about 20-30 degrees. It won't swing shut on its own, nor stay open on its own. It hangs a bit in the middle. Convenient enough to catch you elbow when you're not looking.  I'd like the door to stay put wherever we leave it.  I've heard idea such as slipping cardboard in to the hinge to rebalance the door, etc. but no guidelines as to the thickness of that cardboard, or which hinge to change to move it in which direction.  I would hope you can appreciate how I'd rather not make this a task of trial and error and continually removing the hinge and putting it back on, etc.  I've also heard something about lightly bending one of the hinge pins, but not sure how I should go about that, or "how much" to bend it.  So, I'm looking for some tips on how to fix this (and other) door. The full question: suppose N is normal in G and $[G:N]=p$ is a prime. Prove that for any $K \leq G$, either (1) $K \leq N$ or (2)$KN=G$ and $[K:K \cap N]=p$.  Since $N,K \leq G$, it is of course possible that $K \leq N$ could happen.  but for (2), we can show $KN \subseteq G$  but how can we show $KN=G$?  And what information does "$[G:N]=p$" give us? Encrypt home drive after installation I declined to encrypt to my home drive whilst installing ubuntu.  Is it possible to encrypt it after installation?  if so, how? I've been wondering about this for a while, since getting interested in knife sharpening.  When using a knife, it will get duller. One (the primary) reason being that you "bend" the edge. You fix this using a honing steel.  After a while though, you need to sharpen the knife again.  Another setting: If you have a brittle knife and/or mistreat your knife you will chip it.   As far as I can tell, this should mean that small pieces of metal (dust, shavings, etc.) come lose, and then likely get in the food.  Is this correct? Are there no health considerations when digesting small pieces of very sharp metal? Calculate: $$\int_{0}^{2\pi}e^{R{ {\cos t}}}\cos(R\sin t+3t)\mathrm{d}t$$ My try: $\displaystyle\int_{0}^{2\pi}e^{R{ {\cos t}}}\cos(R\sin t+3t)dt\\ \displaystyle \int_{|z|=R}e^{\mathfrak{R\textrm{z}}}\cos(\mathfrak{I\textrm{z}}+3(-i\log z)dz\\ \displaystyle \int_{|z|=R}e^{\mathfrak{R\textrm{z}}}\mathfrak{R\textrm{e}}^{(\mathfrak{I\textrm{z}}+3(-i\log z))i}dz\\ \displaystyle \int_{|z|=R}e^{\mathfrak{R\textrm{z}}}\mathfrak{R\textrm{e}^{\mathfrak{I\textrm{z}}}}z^{3}dz\\ \displaystyle \int_{|z|=R}e^{\mathfrak{R\textrm{z}}}\mathfrak{R\textrm{e}^{\mathfrak{I\textrm{z}}}R}z^{3}dz$ and there is nothing here that is not holomorphic, therefore according to Cauchy theorem it must be exactly $0$. I want to sort in-text citations by year, but bibliography by name. It sound exactly like .  However, suggested solution (using \usepackage[sorting=ynt]{biblatex} and \printbibliography[sorting=nty]) does not work for me (see MWEB below). Indeed, the list of references remains year-sorted instead of name sorted.  Question: How to force sorting=nty in \printbibliography?      \documentclass{article}     \usepackage[%         backend=biber,%         style=authoryear-comp,%         natbib=true,%         sorting=ynt%     ]{biblatex}     \addbibresource{\jobname.bib}     \usepackage{filecontents}  \begin{filecontents}{\jobname.bib}     @book{key2000,       author = {Author, A.},       year = {2000},       title = {Alphabetical fist \&amp; Year last},       publisher = {Publisher},     }     @book{key1900,         author = {Boathor, B.},         year = {1900},         title = {Alphabetical last \&amp; Year first},         publisher = {Publisher}, } \end{filecontents}  \begin{document}     Citation should be year-sorted, even if \verb+\citep+ is random \citep{key2000, key1900}.% citation NOT year-sorted      \printbibliography[sorting=nyt] \end{document}   P.S.: At the opposite, using  \begingroup     \newrefcontext[sorting=nyt]     \printbibliography \endgroup    overrides the \usepackage[sorting=ynt]{biblatex} option (i.e. LoR is name-sorted, but in-line citations as well). "no public key available" on apt-get update I've often come across posts on forums or other websites where you see people joking in such a manner about running/logging in as root as if it's something awful and everyone ought to know about it. However, there isn't much that a search reveals on the matter.   It may be widely known to Linux experts, but I really don't know why. I remember always running as root when I first tried Linux years ago (Redhat and Mandrake) and don't remember running into any problems because of that.   There are actually some distros that have a bright red background with alert signs all over it as wallpaper for the root user (SuSe?). I still use the "Administrator" account for regular use on my Windows installation and haven't ever run into any problems there either. How can I scale all images in a folder to the same width? I want to scale all images in a given folder to the same width (but different appropriately scaled heights). How can I do this using  a GUI-based or command-line tool?  For bonus points, is it possible to restrict which images are scaled based on their initial width (that is, only scale images that have a width > x and/or a width &lt; y)? A surjective endomorphism (of a Noetherian ring) is injective. The problem is stated as follows:  "Let $R$ be a Noetherian ring and $\theta$ be a ring homomorphism from $R$ to $R$. Show that if $\theta$ is surjective then it is also injective."  Regardless of the right solution, I don't understand why is the following wrong:     We have $\theta: R\to R$. By the isomorphism theorem $R/\ker\theta\cong\operatorname{Im}\theta$. Since $\operatorname{Im}\theta = R$, it follows $\ker\theta =0$, so it's injective.   Harsh criticism will be appreciated. Thanks. how to create a roster sheet in sharepoint 2010 I have IIS7 on my computer, and I can access it through other computers in my house on the same network through my computer's IP address - 192.168.1.10x  Instead of typing in http : //192.168.1.10x/ I want to be able to go to http : //somename/  I know you can change the hosts file to add it but I'd have to do that on each computer and I'd still have to type in the IP to access it from my iPod/Wii... etc. What are the limits on how I can cast, change, and retract votes? What limits are there on how I can vote?  How often can I vote? How do my votes become locked/unlocked? How to access a clicked URL in a URL handler application created in Automator I'm currently trying to modify the default URL handling behavior in OS X. I would like, when clicking on any link outside of a web browser, to copy this URL to the pasteboard instead of opening it in the default web browser – for instance, clicking a link in a Skype conversation should copy this URL to the OS X pasteboard instead of opening it in Firefox (which is currently my default web browser).  I have thought about solutions, but I’m currently unable to implement one. If I can create an application that puts URLs into the pasteboard that are sent to it, and can set this application as the default web browser, I would be able to achieve my goal.  I created an application with Automator that executes the shell command pbcopy and have set this application as my default web browser, but I didn’t get the expected result. I did some testing and I can confirm that the application runs on clicking on a link, however, I cannot get the URL passed to the application / shell script. I tried both stdin or a passed argument, but without success.  Is there a way for an application created with Automator to retrieve a clicked URL and send it to pbcopy, when that application is set as a (default) handler for URLs?  Alternatively, is there a better way to create a fake web browser that takes the input and puts it in the pasteboard than Automator? Or is there any other way that I can achieve my goal without a custom application / script? How do I install gfortran? I am trying to install gfortran in my computer with Ubuntu 12.04 by using these commands,  sudo -i apt-get update apt-get install gfortran   but I found the program saying "unable to locate package gfortran". Then on a Ubuntu page I found some question regarding the same problem and I went through the solution from them but again I couldn't install the gfortran package.  Can anybody tell me how to install it? Social network login or not? I am developing a niche social networking iPhone app of my own.   I was wondering if it would be best to allow the user to connect to social networks such as Facebook and Twitter to login to my app (and of course an eMail/Password fallback) or just go with regular eMail and Password login.  Any thoughts would be appreciated! :) How can I make a message appears on my PC when my phone is ringing? When I'm at work, I usually have a headset on my head and I am listening to music. My phone is always on vibrate to respect co-workers.  However, I don't keep the phone in my pocket because of its size, so I don't feel it vibrating.  I miss a lot of calls.  I was wondering if it's possible to forward "phone is ringing" notifications to the PC, ideally via a Bluetooth connection.  I would not mind installing a 3rd-party app on the phone or software on the PC. The thing with these tags is, you can always (or almost always) use a specific product/technology/language tag which would render the company-name tag completely redundant (see Appendix C for sloppy non-proof). For example, what relevant information would  tag add to  already tagged with ? What would  tag add to  already tagged with ? Another way of looking at this: how much useful information do you lose if you don't tag a question with a company name? Note: I'm not talking about brand/company/organizations like , , ,  because those names also refer to websites/web services with which you can interact via API or from which you can extract data. Appendix C: The &quot;about&quot; test The relationship between a tag and a question can be expressed as such  This question is about [tag]  In case of ,  This question is about [microsoft]  Fair enough. But in reality people are using these company-name tags like this  This question is about something developed by or related to [microsoft]  But if that's the case, you might as well just go with  This question is about [something developed by or related to microsoft]  where  can be , , ... you get the idea. How could the human race be saved in Interstellar? In  (2014) the human race seemed to be facing pretty poor odds until they discovered a convenient wormhole placed by an unknown entity expressed as "they". Later in the movie, we discover that       "they" are actually the human race in the future, reaching back in  time to save itself.   The end of the movie gets a little wild on the physics end, and I'm happy to take that ending as is - no need for it to make sense - but one very basic detail does bother me:  How could the human race have gone back in time to save itself from circumstances that would have caused us to go extinct without our own help (the convenient wormhole)? forecast using arima models I am trying to predict values using arima(0,1,1). After doing predict(mod,n.ahead=5) (in R) am getting the same value for all the predictions:   5947.681 5947.681 5947.681 5947.681 5947.681    Is it correct?