Suppose $\mathbf{X, Y}$ are independent random vectors. Are their components independent? Let $\mathbf{X} = (X_1, \dots, X_p)^\top$ and $\mathbf{Y} = (Y_1, \dots, Y_p)^\top$ be independent. Does it then follow that $X_i$ is independent with $Y_j$ i.e. cov$(X_i, Y_j) = 0$?
Functions of Independent Random Variables 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.
I want to have a page-filling and -breaking longtable, so I decided to try xltabular. Unfortunately xltabular seems to screw up the booktabs rules in the normal tabular environments, see:  \documentclass{article} \usepackage{booktabs, xltabular} \usepackage[active,tightpage]{preview} \PreviewEnvironment{tabular} \begin{document} \begin{tabular}{ccc}     \toprule     col1 &amp; col2 &amp; col3 \\     \midrule     row 2: &amp; more and moooooooooooore &amp; test \\     \bottomrule \end{tabular} \end{document}     Not loading xltabular resolves the issue.  Did I miss some package option or some incompatibilities? Is there a better package than xltabular which doesn't break booktabs?  (I am using TeXLive 2019 and compile with lualatex)    EDIT (by Skillmon)  I reduced the bug to the following MWE (which doesn't make much sense on its own), it looks like booktabs tries to be smart detecting something, but fails badly.  \documentclass{article} \usepackage{longtable} \let\xltabular\relax \usepackage{booktabs}  \begin{document} \begin{tabular}{ccc}     \toprule     col1 &amp; col2 &amp; col3 \\     \midrule     row 2: &amp; more and moooooooooooore &amp; test \\     \bottomrule \end{tabular} \end{document}   It seems to suffice that longtable is loaded and \xltabular is defined (even if it's just \relax). In Ubuntu ISO MBR, why does a GPT partition overlap the boot partition, and make MBR uneditable? How do I use the Natural Earth SQLite DB with QGIS? I just downloaded the Natural Earth data in SQLite format from . I had assumed that this would be a SpatiaLite database, but it appears not to be! QGIS is failing to recognize it as a spatial database. OGR supposedly supports reading geometries even when stored in a plain SQLite database, but maybe QGIS doesn't use OGR for SpatiaLite?  The SQLite database does have geometry_columns and spatial_ref_sys tables. Is there some way to convert it to a full SpatiaLite database? I'm going to submit form using cURL, where some of the contents is come from other file, selected using sed  If param1 is line matching pattern from other file using sed, below command will works fine:  curl -d param1="$(sed -n '/matchpattern/p' file.txt)" -d param2=value2 http://example.com/submit   Now, go to the problem. I want show only text between 2 matching pattern excluding the matching pattern itself.  Lets say file.txt contains:  Bla bla bla firstmatch It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. secondmatch The point of using Lorem Ipsum is that it has a more-or-less normal distribution of letters, as opposed to using 'Content here, content here', making it look like readable English.   Currently, lots of "beetween 2 matching pattern" sed command won't remove firstmatch and secondmatch.  I want the result to become:  It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. Finding $E(|X_1|\ |\max |X_i| )$ Suppose $X_1,\ldots,X_n$ are a random sample of $U(-\theta,\theta)$ with $\theta&gt;0$. How can find $$ E\left(|X_1|\ \big|\max |X_i| \right) $$ that $1\leq i\leq n$? LibreOffice problem while openning a document My problem is that when I try to open one specific .odt document, I can't move the page or do anything except moving the mouse, which is useless. Also, the mouse, when I'm on the document, is freaking out switching between mouse and loader couple of times per second. It also becomes dark when I click on the document like if it's not responding. Highlight tags corresponding to our favorites Each of us have our list of favorite tags.  When a question has tag(s) that match one or more of ours, the background gets highlighted, drawing our attention to the question.  It would also be nice to highlight (perhaps using a bold font) the tags(s) that matched our own, so one could quickly do a further filtering of the question to see if it is of interest at the present time, like the following:    e.g. arduino is in my list of favorites, but the other two in each question are not. My friend had a  on the board with 10 counters on it.  If I destroy it and he sacs it in response to gain the life, is there anything that can be done about that?  Like, can I put another destroy artifact on the stack to resolve before his sac resolves? People are born and grow up and die of old age in a matter of 5 days. The elders (5 days old) train children and send the most promising one on a trek across the dessert to a monolith that is the cause of the rapid aging to hopefully disable it.   I saw it on a reel to reel in the mid 90's. It looks like it was made in the late 70's or 80's. The cast was small. A few men played the elders seated in a row of chairs and the young boy who had to set out to save them who matures into a man and finally an elderly man.  I've been trying to figure out the title for years. Einstein-Hilbert action as an effective field theory I have read the following statement:      "The Einstein-Hilbert action can be seen as an effective field theory: setting $g_{\mu\nu}= \eta_{\mu\nu}+h_{\mu\nu}$, one gets a free Lagrangian for the field $h_{\mu\nu}$ plus interactions."   Now, how is it possible to define all the interaction terms?  EDIT: I ask apologize if my statement was appeared unclear: I have created a  which is more precise; please don't continue to consider seriously this one. Can a character escape the  (DMG p. 177, listed in the SRD as the ) with the  spell? To clarify, I am not asking if the bands prevent teleportation. I am asking whether the bands are teleported with the character as he teleports using misty step. Convert from a specifc range to another one I have installed Ubuntu on my laptop and now I want to remove it and the Grub Loader. How I can do it without the Windows installation disk (I cannot use the Recovery Console). What does “{} \;” mean in find (in Linux)? Sometimes I see a command like  find . -name * -exec ls -a {} \;   I was asked to execute this.  What does {} \; mean here? I use to run  $s =~ s/[^[:print:]]//g;   on Perl to get rid of non printable characters.   In Python there's no POSIX regex classes, and I can't write [:print:] having it mean what I want. I know of no way in Python to detect if a character is printable or not.   What would you do?   EDIT: It has to support Unicode characters as well. The string.printable way will happily strip them out of the output.  curses.ascii.isprint will return false for any unicode character. phpmyadmin not working due to missing extensions I am about to use phpmyadmin on Ubuntu 16.04. But it shows that the page is not working. the error log says error 500   PHP Fatal error:  require_once(): Failed opening required '/usr/share/php/php-gettext/gettext.inc' (include_path='.:/usr/share/php') in /usr/share/phpmyadmin/libraries/select_lang.lib.php on line 477   At first it says that the mysqli|mysql extension is missing. but after sleepless hours I forget what I've been done to it. I am using php7. Can I define keyboard shortcuts using the Super key? For a graph $G$, let $W$ be the (random) vertex occupied at the ﬁrst time the random walk has visited every vertex. That is, $W$ is the last new vertex to be visited by the random walk. Prove the following remarkable fact:      For the   random walk on an $n$-cycle, $W$ is uniformly distributed over all vertices different   from the starting vertex. Expansion of $(1+px+qx^2)^8\equiv 1+8x+52x^2+kx^3$... $(1+px+qx^2)^8 \equiv  1+8x+52x^2+kx^3...$  Given that expression, calculate the values of p, q, and k.  I'm guessing this is a trinomial that needs to be turned into a binomial, but how would you do it? I have no idea. 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?