Let $(X,Y)$ a bivariate normal variable $f(x,y)=\frac{1}{2\pi \sqrt{1-\rho^2}}e^{\frac{1}{2(1-\rho^2)}(x^2-2\rho xy+y^2)}$. Let $Z=\frac{Y-\rho X}{\sqrt{1-\rho^2}}$.  Find the law of $Z$.  Say if $X$ and $Z$ are independent or not.  Find $\mathbb{P}(X&gt;0,Y&gt;0)$.    Already by eye it can be noticed that the mean vector is $\mu=\begin{bmatrix} 0\\ 0\end{bmatrix} $ and the covariance matrix is $\sum=\begin{bmatrix}1 &amp; -\rho\\ -\rho &amp; 1\end{bmatrix}$  . I also know that $Z=X_1+...+X_n\sim N_n(A\mu,A^T\sum A)$, thus $Z$ is a linear of $X$ and $Y$ with $\frac{1}{\sqrt{1-\rho^2}}=const.$. Can I say that $Z\sim N(0,2-2\rho)$?
Given   $$ f_{XY}(x,y) = \frac{1}{2\pi \sqrt{1-\rho^2}} \exp \left( -\frac{x^2 +y^2 - 2\rho xy}{2(1-\rho^2)} \right) $$  $Y = Z\sqrt{1-\rho^2} + \rho X$  And   $$ f_{XZ}(x,z) = \frac{1}{2\pi } \exp \left( -\frac{x^2 +z^2}{2} \right) $$  Show that $P(X&gt;0,Y&gt;0)= \frac{1}{4}+\frac{1}{2\pi}(\arcsin \rho) $  I'm supposed to use the fact that X and Z are independent standard normal random variables, but I don't quite understand how. Any help would be greatly appreciated.
Could Tom Bombadil have defeated Sauron? Well, just another idea came up into my mind and i have no idea how to solve it :D Is there infinitely many prime numbers, which are not repunits and their inverse is also prime? (For example, inverse of 31 is 13 which is also prime. i didn't have any other name to describe the function!) P.S:I now know they are called s. I'd like to update the label of an operator when it's called. But this :  def execute(self,context):     self.bl_label = "good night"   ... apparently doesn't work if the bl_label property has already been defined. Am I wrong ? Properly bootstrap drupal for standalone PHP file? This is what I have right now:  &lt;?php     define('DRUPAL_ROOT', '/var/www/clients/client1/web15/web');     include_once(DRUPAL_ROOT . '/includes/bootstrap.inc');     drupal_bootstrap(DRUPAL_BOOTSTRAP_FULL); //  drupal_flush_all_caches();        $users = views_get_view_result('userlist', 'uid_list');     $i = 0;     foreach ($users as $user) {         $filename = 'sites/default/files/generated_pdfs/individual_pdf_report_' . $user-&gt;uid . '.pdf';         generate_employee_pdf($user-&gt;uid, $filename);         error_log('Generated ' . $filepath);         $i++;     }     // create object     $zip = new ZipArchive();     // open archive     if ($zip-&gt;open("sites/default/files/generated_zips/individual-reports.zip", ZIPARCHIVE::CREATE) !== TRUE) {         die ("Could not open archive");     }     // initialize an iterator     // pass it the directory to be processed     $iterator = new RecursiveIteratorIterator(new RecursiveDirectoryIterator("sites/default/files/generated_pdfs/"));     // iterate over the directory     // add each file found to the archive     foreach ($iterator as $key=&gt;$value) {         $zip-&gt;addFile(realpath($key), $value-&gt;getFilename()) or die ("ERROR: Could not add file: $key");     }     // close and save archive     $zip-&gt;close(); ?&gt;   I'm getting a bunch of errors from the above when run standalone. Works fine in my module. Some of the errors are:  Warning: Invalid argument supplied for foreach() in drupal_depth_first_search() (line 56 of /var/www/clients/client1/web15/web/includes/graph.inc). Warning: Invalid argument supplied for foreach() in _module_build_dependencies() (line 229 of /var/www/clients/client1/web15/web/includes/module.inc). Warning: include_once(sites/all/modules/views_bulk_operations/actions/archive.action.inc): failed to open stream: No such file or directory in views_bulk_operations_load_action_includes() (line 56 of /var/www/clients/client1/web15/web/sites/all/modules/views_bulk_operations/views_bulk_operations.module). Email SMTP validator I need to send hundreds of newsletters, but would like to check first if email exists on server. It's called  validation, at least I think so, based on my research on Internet.    There's several libraries that can do that, and also a page with open-source code in  (), but I have hard time reading ASP Classic, and it seems that it uses some third-party library...    Is there some code for SMTP validation in C#, and/or general explanation of how it works? What's the cleanest, most effective way to validate decimal numbers in JavaScript?  Bonus points for:   Clarity. Solution should be clean and simple. Cross-platform.   Test cases:  01. IsNumeric('-1')      =&gt; true 02. IsNumeric('-1.5')    =&gt; true 03. IsNumeric('0')       =&gt; true 04. IsNumeric('0.42')    =&gt; true 05. IsNumeric('.42')     =&gt; true 06. IsNumeric('99,999')  =&gt; false 07. IsNumeric('0x89f')   =&gt; false 08. IsNumeric('#abcdef') =&gt; false 09. IsNumeric('1.2.3')   =&gt; false 10. IsNumeric('')        =&gt; false 11. IsNumeric('blah')    =&gt; false Are  and the  really the same thing?  Not too long ago, Susskind and Bousso uploaded the article "" with the thesis that the many-worlds interpretation and the multiverse of eternal inflation are one and the same thing. The parallel worlds of one are exactly the same thing as the parallel worlds of the other.   First, they claim decoherence can't happen over a complete description of the future light-cone of the measurement. Then, they apply that principle to eternal inflation. Without decoherence, superpositions of nucleating bubbles and metastable vacua can't decohere. According to the anthropic principle, most bubbles have no conscious observers, but an exponentially small minority do. Apply black hole complementarity to causal horizons.  Then, somehow, in a way I can't follow, they combine causal diamond worlds into a global multiverse.  Then they claim decoherence is reversible.  My head is spinning. What are your opinions on this paper? Which answer do I accept if I have multiple correct answers? What does goodness of fit mean in context of linear regression I am having trouble understanding the concept of 'goodness of fit' w.r.t linear regression. The name suggests that the goodness of fit test is used to determine how well a model fits the data.  But the linear model is already the 'best' fit since we have minimized the sum of the squared error terms. Why do we need to test 'goodness of fit' again? We know it is a good fit because we minimized the sum of the squared errors. Do not delete good duplicates! WPF textblock binding with List does anyone know if there is a simple way to bind a textblock to a List. What I've done so far is create a listview and bind it to the List and then I have a template within the listview that uses a single textblock.  what I'd really like to do is just bind the List to a textblock and have it display all the lines.  In Winforms there was a "Lines" property that I could just throw the List into, but I'm not seeing it on the WPF textblock, or TextBox.  Any ideas?  did I miss something simple?  Here's the code  &lt;UserControl x:Class="QSTClient.Infrastructure.Library.Views.WorkItemLogView" xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation" xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"          Width="500" Height="400"&gt; &lt;StackPanel&gt;     &lt;ListView ItemsSource="{Binding Path=Logs}" &gt;         &lt;ListView.View&gt;             &lt;GridView&gt;                 &lt;GridViewColumn Header="Log Message"&gt;                     &lt;GridViewColumn.CellTemplate&gt;                         &lt;DataTemplate&gt;                             &lt;TextBlock Text="{Binding}"/&gt;                         &lt;/DataTemplate&gt;                     &lt;/GridViewColumn.CellTemplate&gt;                 &lt;/GridViewColumn&gt;             &lt;/GridView&gt;         &lt;/ListView.View&gt;     &lt;/ListView&gt; &lt;/StackPanel&gt;     and the WorkItem Class  public class WorkItem {     public string Name { get; set; }     public string Description { get; set; }     public string CurrentLog { get; private set; }     public string CurrentStatus { get; private set; }     public WorkItemStatus Status { get; set; }     public ThreadSafeObservableCollection&lt;string&gt; Logs{get;private set;}   I'm using Prism to create the control and put it into a WindowRegion          WorkItemLogView newView = container.Resolve&lt;WorkItemLogView&gt;();         newView.DataContext = workItem;         regionManager.Regions["ShellWindowRegion"].Add(newView);   thanks When I'm using  find . -type f -name "*.htm*" -o -name "*.js*" -o -name "*.txt"  it finds all the types of file. But when I add -exec at the end:  find . -type f -name "*.htm*" -o -name "*.js*" -o -name "*.txt" -exec sh -c 'echo "$0"' {} \;  it seems it only prints .txt files. What am I doing wrong?  Note: using MINGW (Git Bash) How do I kill entities at y=65? Potions can survive at y=65  One potion was thrown into the void, but instead of being killed, it just lingered there.  Any ideas to kill it? I can't just run /kill @e[type=!player,tag=test1]  (yes, the tag of the entity is indeed test1 ) Blender 2.80 AV Sync missing? I just installed Blender 2.8 and for the life of me I can not find AV Sync anywhere and video editing has become impossible. Is there some new setting I am unaware of? Get the index of the nth occurrence of a string? Until I went to army, I (had been / used to be / was ) a introverted person. - which one is right? Until I went into the army, I (had been / used to be / was ) an introverted person. And after I I finished army service, I became more confident.   I think all of these are possible.  As for had been, I'm referring to the state of me being introverted before the accident that happened in the past. So, I think I can use had been.  And at the same time I feel I can use used to be and was.. If I don't want to emphasize the time gap between two accident, I feel I don't need to use past perfect. So I think this problem depends on how I'm going to say something.   which one is preferable? and what is the difference? Is there any way to tell Google to display the updated title instead of displaying the old blog title in search results? As a matter of optimization, a few weeks ago I modified the way the titles of my blog posts are displayed (I made a change also in the title of the blog). Then I used the Google Webmaster Tools to re-index the old pages and update titles. Immediately I could update the titles and posts appear correctly on Google.  The problem is that now the titles of posts began to appear incorrectly in the search, using the old blog title. Some posts continue to display the updated title, but many recent posts show the old title.  I've tried re-index the pages with Google Webmaster Tools, but the titles remain unchanged in the search results.  Is there any way to "force" Google to display the correct titles? Difference between `dpkg -i` and `apt-get install` From all the other Q&amp;A (like: ) I understand that apt-get is a wrapper over dpkg and provides dependency management.  Assuming I'm installing something that doesn't have any dependencies (or they're also installed at the same time) would there be any difference between installing it using apt-get vs dpkg  E.g. (kinda concrete differences I'm looking for)   say dpkg can not download the pkg from somewhere, I must download before starting installation, whereas apt-get will do this on my behalf. dpkg won't configure pkg, it'll only copy files. apt-get will configure too.   ... Does timbre consist in pitch and volume? I read that the physical properties of a sound wave correspond to its audible qualities: pitch, volume, and timbre. However, an oscilloscope uses only two-dimensions to accurately depict the physical properties of a wave. Intuitively, pitch and volume seem more basic than timbre does, so I surmise that timbre must consist in those two properties.    Does timbre consist in those two properties? How do two-dimensional waveform diagrams depict three properties of sound waves? Deriving curvature formula How do you derive the formula for unsigned curvature of a curve $\gamma (t) = (x(t), y(t)$ which is not necessarily parameterised by arc-length.   The formula given is $$|\kappa (t)| = \left|\frac{x'' y' - y'' x'}{((x')^2 + (y')^2)^{3/2}}\right|$$  All the definitions of curvature in my notes give it in terms for an arc-length parameterisation $T'(s) = \kappa(s) N(s)$. This is apparently meant to be 'easily derivable', am I missing something?