Consider random variable $X$ with continuous, increasing CDF $F_X (x)$. Let $W=F_X (X)$. Characterize the distribution of $W$.  I get $F_W (w)=\mathrm{Pr}(F_X (X) \leq w)=\mathrm{Pr}(X\leq F_X^{-1}(w))=F_X(F_X^{-1}(w))=w$  Can I just impose that $w=1$ and say this is a degenerate distribution? That seems off to me because the question doesn't actually specify a domain for $x$. Did I miss something?
For a CDF that is not strictly increasing, i.e. its inverse is not defined, define the quantile function   $$F^{-1} (u) =\inf \{x: F(x) \geq u \},\quad 0&lt;u&lt;1. $$  Where U has a uniform $(0,1)$ distribution. Prove that the random variable $F^{-1} (u)$ has cdf $F(x)$.  In case of a strictly increasing CDF the proof is quite easy because the inverse is defined. Define $X=F^{-1} (u)$  $$ P\left[X&lt;x \right]=  P \left [F^{-1} (U) \leq x \right]= P \left[U \leq F(x) \right] =F(x)  $$  But how do I accomondate the nondecreasing CDF whose inverse is given by the quantile function? I am a begginer so any help is welcome. Thank you.
At what time intervals does ntpd update the time? I'm running the latest ntpd.   When I start ntpd my system time synchronizes with ntp server's ntp. After synchronizing I changed my system time manually using date command date -s '1997-02-22 12:00:00'   My system time got changed as per the date command.   NTPD is still running, I want to know at what time interval my system time will sync with the internet via ntp. "A mice problem" vs. "a mouse problem" Extended Euclidean Algorithm: backward vs. forward How much longer will my 10 year old frame last? How to reduce the spacing between responses in the alterqcm package? This question has a continuation here   The alterqcm package is no longer maintained by Alain Matthes. Two other developers maintain it. I wrote to them 15 days ago by email to report the error reported here:  I have not received a reply. I am trying to reduce the spacing between the lines of the QCM as indicated in  by @Werner and it does not work. In addition, by using the David Carlisle's code given in , I get an X instead of a square.  Full code of the screenshot: \documentclass[12pt]{article} %\usepackage[utf8]{inputenc} %\usepackage[upright]{fourier} \usepackage{etoolbox}% http://ctan.org/pkg/etoolbox \usepackage[english]{alterqcm} \def\square{X} % &lt;---------- David Carlisle comment \def\nogreekalph{??} % &lt;---- David Carlisle comment \parindent0pt \begin{document} % Original display \begin{alterqcm}[lq=8cm]  \AQquestion{Question}{%  {Proposition 1},  {Proposition 2},  {Proposition 3}} \end{alterqcm}  \bigskip  % Smashed display !!!!!! Dont work now  {\begin{alterqcm}[lq=8cm]  \AQquestion{Question}{%  {\smash{Proposition 1}},  {\smash{Proposition 2}},  {\smash{Proposition 3}}} \end{alterqcm}}  \bigskip  % Global smashed display !!!! Dont work now  \makeatletter \patchcmd{\aq@prop}{\vbox{#1}}{\vbox{\smash{#1}}}{}{}% patch \aq@prop \makeatother \begin{alterqcm}[lq=8cm]  \AQquestion{Question}{%  {Proposition 1},  {Proposition 2},  {Proposition 3}} \end{alterqcm}  \end{document}  Is there a solution? When adjusting the volume in Windows 8.1, a box appears at the top left corner of the screen. This box sticks around for ~13 seconds before it fades away. If you attempt to dismiss it by click on it, it switches to the music app.  How do you disable or dismiss this annoying box? At the very least, can the time visible be adjusted? More times than not, it's in the way. Does the Rule of Two Serve any Actual Purpose? I know Darth Bane came up with the Rule of Two to ensure the survival of the Sith, but, in reality (or in the Star Wars reality), does it (or any Sith rule) serve a true purpose?  And why does each successive Sith bother with it?  Due to the nature of the Sith, Sith Lords tend toward arrogance and self-importance.  Both the Sith master and Sith apprentice seem to regularly break the rule by training others.  Deceit and personal interest over loyalty is a way of life for the Sith.  What is to keep any Sith from following the Rule of Two (or any rule, for that matter) if it is against their own self interest?  If a Sith Master can stay in power longer by training a 2nd apprentice and pitting the two against each other, is he really going to worry about the Rule of Two more than his own plans and survival?  The Rule of Two also assumes that every Sith Master would place the survival of the Sith over his own survival, as opposed to attempting to ensure his own immortality in whatever way he could (like Plagueis).  When you have a group that, by default, is arrogant and self-serving, as well as deceptive, why would they want the Sith to survive themselves?  I can see how the Rule of Two, and any other Sith beliefs would be possible guidelines, but it terms of the nature of ambitious people (and the Sith are ambitious), it's hard to believe that each new Sith Master would follow this rule faithfully.  So does this rule, or any Sith rule, actually work?  Or is it more of a suggestion that is broken when convenient? I'd like to have a python program alert me when it has completed its task by making a beep noise.  Currently,  I use import os and then use a command line speech program to say &quot;Process complete&quot;.  I much rather it be a simple &quot;bell.&quot; I know that there's a function that can be used in Cocoa apps, NSBeep, but I don't think that has much anything to do with this. I've also tried print(\a)  but that didn't work. I'm using a Mac, if you couldn't tell by my Cocoa comment, so that may help. OK we are all adults here, so really, how on earth should I live without bidet? This is the part four in the interesting series: "OK we are all X here, so really". You can find the past episodes here:        Travelling always requires a bit of adjustment. While doing our favorite activity we always leave home our comfort, our routines and, unfortunately, Italians (and others) leave home their beloved .  The question is simple: what are the substitutes / tips&amp;tricks for bidet while travelling in bidet-unfriendly countries so that we can do our business without end up feeling, let's say, uncomfortable? Syntax highlighting in app differs from normal web view How can I solve this integral using the Residue theorem? $$ \int_{-\infty}^\infty \frac{\sin(x)}{x^2 + 4x + 5}dx$$  What I now is you can use residue theorem, and evaluate this integral  $$ \oint \frac{e^{iz}}{z^2 + 4z + 5}dz$$ and through calculating the residues and using the residue theorem I get that the answer for the above integral is $$\pi e^{-1-2i}$$  However after this I am completely lost.  Any guidance will be much appreciated. I would like to know why I can evaluate the integral as an exponential integral as well. From my understanding it's not prime because it's not greater than $0$. So my followup question is why did mathematicians exclude $-1$?  The definition of prime is having only two factors.   $-1 \cdot 1 = -1$ Instead of the constrained 45 degrees that happens when you press C, is there a specific angle I can get for example 35 degrees instead?   It's currently cut at 45 degrees in the attached image and I need it to be 35. Is the EmDrive, or "Relativity Drive" possible? In 2006, New Scientist magazine published an article titled 1 [] about the  [] which stirred up a fair degree of controversy and  that New Scientist was engaging in pseudo-science.   Since the original article the inventor claims that a "Technology Transfer contract with a major US aerospace company was successfully completed", and that papers have been published by Professor Yang Juan of The North Western Polytechnical University, Xi'an, China. 2   Furthermore, it was  that the Chinese were going to attempt to build the device.  Assuming that the inventor is operating in good faith and that the device actually works, is there another explanation of the claimed resulting propulsion?  Notes:  1. Direct links to the article may not work as it seems to have been archived. 2. The abstracts provided on the  website claim that they are Chinese language journals which makes them very difficult to chase down and verify. Rotation of braces How can I rotate a brace to make it appear ?  I'm currently using this code:  \documentclass[french]{beamer} \usepackage{tikz} \usetikzlibrary{fit,calc,positioning,decorations.pathreplacing,matrix} \begin{document} \begin{frame} \begin{tikzpicture}[decoration={brace}][scale=2]  \node [rotate=270] (A) {\fontsize{8}{8}\selectfont A};  \node [fit=(A)] (fit) {};               \draw [decorate,line width=1pt]    ([yshift=7pt]fit.south west) -- ([yshift=-7pt]fit.north west); \end{tikzpicture} \end{frame} \end{document}   Thanks in advance! When installing an application, the application lists permissions that it needs to perform its  functions.    I am creating this list of the the system defined permissions and a description of what they mean. It is a community wiki so if new permissions are added in the future they can be added to this list. How to use "who" vs. "that" I often get confused when trying to use who vs that.  Some examples that often confuse me:  That     The person that went to the store.      The people that went shopping.      The persons that went shopping.      The group that went shopping.   Who     The person who went to the store.      The people who go shopping.   Please explain when to use either for plural subjects and singular subjects.  Animate and inanimate objects as well. Collapsing a stream of multiple lines into one line I have a file containing a a large number of lines, each of which contains a bunch of numbers which are separated by spaces. I process this data in a pipe in some way, and then I want to collapse the multiple lines into a single line of all the numbers separated by spaces.  Is there a standard command-line utility I can use to do this? It seems like most line-by-line utilities won't mess with the newlines... Get PHP to stop replacing '.' characters in $_GET or $_POST arrays? If I pass PHP variables with . in their names via $_GET PHP auto-replaces them with _ characters. For example:  &lt;?php echo "url is ".$_SERVER['REQUEST_URI']."&lt;p&gt;"; echo "x.y is ".$_GET['x.y'].".&lt;p&gt;"; echo "x_y is ".$_GET['x_y'].".&lt;p&gt;";   ... outputs the following:  url is /SpShipTool/php/testGetUrl.php?x.y=a.b x.y is . x_y is a.b.   ... my question is this: is there any way I can get this to stop? Cannot for the life of me figure out what I've done to deserve this  PHP version I'm running with is 5.2.4-2ubuntu5.3. How do I install and manage GNOME Shell extensions? How do I install GNOME Shell extensions?  Is it possible to update, remove and manage the extensions?