Distribution function of a random variable defined by a cumulative distribution function
Proof of $Y=F_X(X)$ being uniformly distributed on $[0,1]$ for arbitrary continuous $F_X$
GTA V for PC exited unexpectedly I recently ordered the GTA V PC from Steam and when I finished installing and launch the game. It gave me "Grand Theft Auto V has exited unexpectedly". I tried "safe mod" and every option available, but none of them work. I checked some forums and found a lot of people having the same problems. What could be the possible cause? I'm using a laptop with these specs:  CPU: i7 4700m GPU: AMD HD8970M RAM: 8GB  I'm pretty sure my laptop is way beyond the minimum specs. Does someone know how can I enter the game? When umpire change the cricket ball, was the old ball thrown as waste or keep for reuse? I'm flying to Tokyo. The flight was sold through Finnair. The first leg is operated by American Airlines. The second and final legs are operated by Finnair.  When I checked in online, it said that some airports on my journey (didn't say which one) don't allow mobile boarding passes?? Meaning I will need a physical copy.  Unfortunately I don't have easy access to a printer. Will I still be able to print all my boarding passes at the airport even though I already checked in online? Evaluate the integral $\int \frac{\sec x}{\sqrt{3+\tan x}}dx$ Evaluate the following integral.$$\int \frac{\sec x}{\sqrt{3+\tan x}}dx$$  On putting $t=\tan x$ I am getting $$\int\frac{1}{\sqrt{(t^2+1)(t+3)}}dt$$.  How should i proceed from here. Creating point features with exact coordinates in QGIS I've been living in Ireland for almost a year now and I start noticing they use the word "mens" a lot. I can see it used in:   Shops, to denote the area where you can find men's clothes In sport, when they talk about "mens team".    My guess is just that they are lazy about the use of quotes, so that mens should actually be men's. However, there may be some rule I'm not aware of. So, is "mens" only limited to Irish English? When I'm allowed to use it? Why isn't Derivative equal to NeumannValue? Prove by mathematical induction that $2n ≤ 2^n$, for all integer $n≥1$? I need to prove  $2n \leq 2^n$, for all integer $n≥1$ by mathematical induction?  This is how I prove this:  Prove:$2n ≤ 2^n$, for all integer $n≥1$  Proof: $2+4+6+...+2n=2^n$  $i.)$ Let $P(n)=1         P(1): 2(1)=2^1\implies 2=2$.        Hence, $P(1)$ is true.  $ii.)$  Assume that $P(n)$ is true for $n=k$, i.e, $2+4+6+...+2k=2^k$, and prove that $P(n)$ is also true for $n=k+1$, i.e, $2+4+6...+2(k+1)=2^{(k+1)}$  from the assumption add $2(k+1)$ on both sides so we have   $2+4+6...2k+2(k+1)=2^k+2(k+1)$  I'm confused with $2^k+2(k+1)$, I don't know how to make $2^k$ be equivalent to $2^{k+1}$.  I feel i'm doing something wrong.   Any help would be appreciated! When changing fields, should I focus more on describing past research or connecting it to the new area in my statement of purpose? I want to change my field from molecular biophysics to theoretical ecology. While they are basically required some same basic background (physics, math, programming), the rest is quite irrelevant. In my current draft of this part, about a half is about this the problem and the result of the work, another half is for my contribution and what I've learnt. Having said that, should I focus less on describing the work and leave space for the thing that will be useful for the next research?  Below is copy from .     What have you already done? What problems have you solved, or at least worked on? What independent projects have you been part of? What were your key contributions? What did you learn? What did you teach the world? How do your results compare to what was already known? What original ideas are you most proud of? Be specific, technical, credible, and confident (but not arrogant). Refer the reader to your web page for more details. Have a web page with more details: preprints, project reports, source code, videos, etc. Set margins in maketitle Is it possible to change the margins in make title, so they match the margins of the document?  I have created the following modification to the \maketitle. I would like everything to align to at the same left margin.    \documentclass[a4paper]{article} %=============================================== % Title and author \makeatletter \def\@maketitle{%   \newpage   \vspace*{-\topskip}      % remove the initial space    \flushleft Incomplete draft. Please do not cite or quote without premission   \vskip 2em   {}   \begingroup\flushleft    % instead of \begin{center}   \let \footnote \thanks   \hrule height \z@        % to avoid the insertion of lineskip glue     {\LARGE \protect\textbf{\@title} \par}%     \vskip 1.5em     {\large       \lineskip .5em          \@author         \par}%     \vskip 1em     {\normalfont       Last updated on \@date}   \par\endgroup            % instead of \end{center}   \vskip 1.5em             % &lt;--- modify this to adjust the separation } \makeatother %============================================== \title{Physician Information Acquisition In a Dynamic Setting} %\email{rud} \author{ \large {\Large Rud Faden} \\ \textit{\normalfont University of Copenhagen} \\ \textit{{\small \href{mailto:rudfaden@gmail.com}{rudfaden@gmail.com}}} } \date{\today} % no date \usepackage{hyperref} \begin{document} \maketitle \abstract{Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enimad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui ocia deserunt mollit anim id est laborum.}    \section{First} Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. \end{document} Is it possible to split micro USB to USB cable so two cables (data + and data -) go to the PC with USB and two other power cables go to power brick? What happens if I activate a product key on Steam for a game I already own? I have a Steam product key for Just Cause 2 I got for free with the purchase of my video card. However, I already have a copy of this game attached to my Steam account. What will happen if I try to activate this other key on my account? Will Steam give me a copy of Just Cause 2 to gift to somebody else? Will it simply refuse to add it? Or will it eat the key rendering it worthless?  I don't want to try this only to have the latter happen, but it would be useful to be able to add it and gift it to somebody else. How to find Oracle's installed version and release from a program? I need to prove that $\mathbb{R}\backslash\mathbb{Q} \sim \mathbb{R} $  Using Cantor-Bernstein, need to show an injection from $\mathbb{R}\backslash\mathbb{Q}$ to $\mathbb{R}$ and from $\mathbb{R}$ to $\mathbb{R}\backslash\mathbb{Q}$.    $\mathbb{R}\backslash\mathbb{Q}$ is a subset of $\mathbb{R}$ so only need to show injection from $\mathbb{R}$ to $\mathbb{R}\backslash\mathbb{Q}$ to complete the proof.    Possible injection: $f:\mathbb{R}\to \mathbb{R}\backslash\mathbb{Q}$ defined as $f(x) = \pi x$ if $x$ is not a multiple of $\pi$; otherwise $f(x) = \sqrt{2} x$.    Not sure if $f$ actually is an injection... Can I have dynamic custom variables in Word? I'm basically trying to create a variable in a document, let's say the name of a person, so that I can change the value in a single place and it updates through the whole document.  I've been reading about Fields and about a DOCVARIABLE, but I can't find where to edit the value (right clic didn't show an "edit" option). I'm using Word 2011 for Mac (that's what we have at work right now) so I'm wondering whether this version of Word is just not intuitive, or I cannot have variables like that in Word and I'm misinterpreting the documentation.  This is what I intend to do:  Hello, my name is PERSON_NAME, I have PERSON_AGE years old.   You get the idea.  PS: We actually want to modify the variables' values from our app, rather than from Word itself, kind of like injecting values to a .*docx file. Can Gmail automatically base "send from" address on the "to" address? For every set $A$, the empty set is a subset of $A$. The empty set is a set. Therefore, the empty set has a cardinality $\geq 1\ldots$ I have only recently been exposed to sets. According to , as seen on the first bullet mark of the link, $ \forall A: \emptyset \subseteq A$  Does this mean that $\emptyset$ is an element of all sets? (This is False, thank you those that answered)  Is the empty set also a set itself?  Assuming these statements are true, then the empty set therefore an element of the empty set. This does not sound right, please clarify for me. Thank you. How can I evaluate a C# expression dynamically? I would like to do the equivalent of:  object result = Eval("1 + 3"); string now    = Eval("System.DateTime.Now().ToString()") as string   Following Biri s , I got this snippet (modified to remove obsolete method ICodeCompiler.CreateCompiler():  private object Eval(string sExpression) {     CSharpCodeProvider c = new CSharpCodeProvider();     CompilerParameters cp = new CompilerParameters();      cp.ReferencedAssemblies.Add("system.dll");      cp.CompilerOptions = "/t:library";     cp.GenerateInMemory = true;      StringBuilder sb = new StringBuilder("");     sb.Append("using System;\n");      sb.Append("namespace CSCodeEvaler{ \n");     sb.Append("public class CSCodeEvaler{ \n");     sb.Append("public object EvalCode(){\n");     sb.Append("return " + sExpression + "; \n");     sb.Append("} \n");     sb.Append("} \n");     sb.Append("}\n");      CompilerResults cr = c.CompileAssemblyFromSource(cp, sb.ToString());     if (cr.Errors.Count &gt; 0)     {         throw new InvalidExpressionException(             string.Format("Error ({0}) evaluating: {1}",              cr.Errors[0].ErrorText, sExpression));     }      System.Reflection.Assembly a = cr.CompiledAssembly;     object o = a.CreateInstance("CSCodeEvaler.CSCodeEvaler");      Type t = o.GetType();     MethodInfo mi = t.GetMethod("EvalCode");      object s = mi.Invoke(o, null);     return s;  } What are good reasons to prohibit inheritance in Java, for example by using final classes or classes using a single, private parameterless constructor? What are good reasons of making a method final? I need help finding the set of continuous functions $f : \Bbb R \to \Bbb R$ such that for all $x \in \Bbb R$, the following integral converges:  $$\int_0^1 \frac {f(x+t) - f(x)} {t^2} \ \mathrm dt$$  I think it might be the set of constant functions but i havent been able to prove it :(   I was thinking that you can use the stone weiestrass theorem considering the set of continuous functions on a closed  interval(non trivial) ,and a subset which contains the set of continuous functions whose integral above diverges in some point in that interval along with  with the set of constant functions. So in order to solve the problem i need only to prove that if two functions do not meet the condition of the problem then their product does not as well .  I hope you can provide some insight and thank you .