Let $(X_{n}), n= 1,...4$ independent real valued random variables. Are $(X_1,X_2)$ and $(X_3,X_4)$ independent? If so please prove why.
Is the claim that functions of independent random variables are themselves independent, true?   I have seen that result often used implicitly in some proofs, for example in the proof of independence between the sample mean and the sample variance of a normal distribution, but I have not been able to find justification for it. It seems that some authors take it as given but I am not certain that this is always the case.
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,, How does comment voting and flagging work? Comments can now be upvoted and flagged. How do those features work? Do users get rep for their upvoted comments? Do upvoted comments have any effect on comment sorting? And can users see their own comment upvote number somewhere?  Related:  The  about comments I frequently find myself writing code like this:  List&lt;int&gt; list = new List&lt;int&gt; { 1, 3, 5 }; foreach (int i in list) {     Console.Write("{0}\t", i.ToString()); } Console.WriteLine();   Better would be something like this:  List&lt;int&gt; list = new List&lt;int&gt; { 1, 3, 5 }; Console.WriteLine("{0}\t", list);   I suspect there's some clever way of doing this, but I don't see it. Does anybody have a better solution than the first block? Extract substring in Bash I'm writing an essay, and want to know the promotion rate and average duration from assistant professor to associate professor and associate professor to professor in recent years. Sadly I don't know the data for the US. Could anyone tell me? How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to , but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method.  In general, how can I evaluate $$\sum_{n=0}^\infty (n+1)x^n?$$ Let's disallow moderator nominations from people who've been suspended in the past year Prove that if every node in a simple graph $G$ has degree $3$ or higher, then $G$ contains a cycle with a chord. I am moving from one job to another (postdoc to tenure track) this summer and I have papers in several stages of being written, these go from handwritten notes with all the necessary arguments to under revision at a journal. All the papers will have my new contact information put on them but where is the line for changing affiliation? It seems obvious to me that the one that is still in the handwritten notes stage will get the new affiliation and that the ones under revision won't. So somewhere in between is a line.   Where is it?  There are some similar questions so far (see  and  but they do not seem to be quite the same situation since I am not changing fields and the appointments are not concurrent. Creating additional Virtual Host Ubuntu Server Can I paste plain text by default? I regularly copy and paste text between spreadsheets, emails, browser windows, etc. I can't think of a single time when I've wanted to keep the formatting from the source text.  I already know about the following workarounds:   Use “Edit” → “Paste Special” → “Unformatted text” Paste to Notepad and copy from there Install a program that takes over the clipboard   What I want is to tell Windows to just do this by default.  Is this possible? If $X$ is compact and $T:X \to Y$ continuous and bijective, show that $T$ is homeomorphism. Let $X$ and $Y$ be metric spaces, $X$ compact, and $T:X \to Y$ bijective and continuous. Show that $T$ is a homeomorphism.   My attempt:  We need only show that $T^{-1}$ is continuous.  Let $M \subset Y$ be closed. Then, from the surjectiveness of $T$, there exists a $A \subset X$ such that $T(A)=M$. Since $M$ is closed, by the continuity of $T$, we must have that $A$ is closed.  Now $$T^{-1}(T(A))=T^{-1}(M)$$ i.e, $A=T^{-1}(M)$. Since $A$ and $M$ are closed, we must have that $T^{-1}$ is continuous.  My problem with this, however, is that I am not using the fact that $X$ is compact anywhere in the proof. Is there perhaps another way that I can show this, using the compactness of $X$?   Any hints will be appreciated. Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups? I am travelling from Singapore to San Francisco and the airline has a layover of 7.5 hours at Tokyo Airport (NRT). I am wondering if I would be allowed to go out to experience the city for a while in some form a visitor visa.  I know Singapore has such facilities, but does anyone know if I can do it in Tokyo? Any suggestions or pointers are welcome. Note - I do not have a visa for Japan yet, but I am hoping that 1 day visitor visa could be possible at airport. New installation – TeX editor says: "Error: Could not start the command" I just downloaded and installed Texmaker. I've never used any (La)TeX before.  When I click the Run button with an arrow next to "Quick build" I get an error saying "Could not start the command". Why? What do I need to do? "Inspirational" quote at start of chapter What is the antiderivative of $e^{-x^2}$ I was wondering what the antiderivative of $e^{-x^2}$ was, and when I wolfram alpha'd it I got $$\displaystyle \int e^{-x^2} \textrm{d}x = \dfrac{1}{2} \sqrt{\pi} \space  \text{erf} (x) + C$$  So, I of course didn't know what this $\text{erf}$ was and I looked it up on wikipedia, where it was defined as:   $$ \text{erf}(x) = \dfrac{2}{\sqrt{\pi}} \displaystyle \int_0^x e^{-t^2} \textrm{d}t $$   To my mathematically illiterate mind, this is a bit too circular to understand. Why can't we express $\int e^{-x^2} \textrm{d}x$ as a 'normal function'? Also, what is the use of the error function? Merge files based on matching of first column I have two files: File1  ARS-BFGL-BAC-10975 0.9303 688423261 1 01/04/2015 0.9983763305 ARS-BFGL-BAC-11025 0.9092 688423261 1 01/04/2015 0.9983763305 ARS-BFGL-BAC-11044 0.9626 688423261 2 01/04/2015 0.9983763305 ARS-BFGL-BAC-11193 0.9544 688423261 1 01/04/2015 0.9983763305 ARS-BFGL-BAC-10975 0.9303 688423263 1 01/04/2015 0.9983763305 ARS-BFGL-BAC-11025 0.9092 688423263 1 01/04/2015 0.9983763305 ARS-BFGL-BAC-11044 0.9626 688423263 2 01/04/2015 0.9983763305 ARS-BFGL-BAC-11193 0.9544 688423263 1 01/04/2015 0.9983763305   File2:  ARS-BFGL-BAC-10975 10 21225382 ARS-BFGL-BAC-11025 10 84516867 ARS-BFGL-BAC-11193 1 29303546   Desired output  ARS-BFGL-BAC-10975 0.9303 688423261 1 01/04/2015 0.9983763305 10 21225382 ARS-BFGL-BAC-11025 0.9092 688423261 1 01/04/2015 0.9983763305 10 84516867 ARS-BFGL-BAC-11193 0.9544 688423261 1 01/04/2015 0.9983763305 1 29303546 ARS-BFGL-BAC-10975 0.9303 688423263 1 01/04/2015 0.9983763305 10 21225382 ARS-BFGL-BAC-11025 0.9092 688423263 1 01/04/2015 0.9983763305 10 84516867 ARS-BFGL-BAC-11193 0.9544 688423263 1 01/04/2015 0.9983763305 1 29303546   So file 1 has many more rows than file 2. I only want to keep rows in the output that are in file 2 based on column1.   I have tried join but I cant get it to work right-it will tell me my files are not sorted  join -j 1 -o 1.1,1.2,1.3,1.4,1.5,1.6,2.2,2.3 &lt;(sort -k1 file1) &lt;(sort -k1 file2)   Preferably I would prefer an awk command. File 1 will be very large. I have tried   awk 'FNR==NR{a[$1]=$2 FS $3;next}{ print $0, a[$1]}' file2 file1 &gt; output   Any help would be much appreciated. Thanks  Sorry I cant comment below but just to clarify file not all rows in column1 in file 1 will be in file2.  The awk command  awk 'FNR==NR{a[$1]=$2 FS $3;next} $1 in a {print $0, a[$1]}'    will only keep the amount of rows there is in file 2. But ideally what I want is were for expample ARS-BFGL-10975 is repeated twice (realistically way more) to appear twice in my output.  Thanks for the help so far Since the Denoise feature was added to Blender I only use the default setting with it and it really does the work. But sometimes the denoise is noticable on some of my renders. How can I overcome the blurred effect? Denoising for me is a big help because my system is quite aged. I have 2-3 different column names that I want to look up in the entire DB and list out all tables which have those columns. Any easy script?