Can someone help me give me some pointers as to how to prove this relation?
Let $X$ be a non-negative random variable and $F_{X}$ the corresponding CDF. Show,   $$E(X) = \int_0^\infty (1-F_X (t)) \, dt$$   when $X$ has : a) a discrete distribution, b) a continuous distribution.   I assumed that for the case of a continuous distribution, since $F_X (t) = \mathbb{P}(X\leq t)$, then $1-F_X (t) = 1- \mathbb{P}(X\leq t) = \mathbb{P}(X&gt;  t)$. Although how useful integrating that is, I really have no idea.
What do the symbols displayed by ls -F mean? I noticed that if I run ls -F on a directory, some of the entries have a * or a @ after them.   spuder@ubuntu:~$ ls -F /sbin acpi_available*   getpcaps*           lvmconf*                 ntfscp*        start-stop-daemon* agetty*           getty*              lvmdiskscan@             ntfslabel*     status@ alsa*             halt@               lvmdump*                 ntfsresize*    stop@ alsactl*          hdparm*             lvmsadc@      spuder@ubuntu:~$ ls -F ~ daq-0.6.1/  examples.desktop       noname-cache.lib  snort-2.9.1/   Templates/ Desktop/    jpgraph-1.27.1/        noname.sch        snortfiles/    Ubuntu One/ Documents/     According to the ls man pages   spuder@ubuntu:~$ man ls ... -F, --classify   append indicator (one of */=&gt;@|) to entries ...   I'm guessing that @ means symbolic link,     What do these other indicators mean ( */=&gt;@| )  ? Evaluate : $1+\frac{2}{6}+\frac{2\cdot 5}{6\cdot 12}+\frac{2\cdot 5\cdot 8}{6\cdot12\cdot 18}+\ldots$ Evaluate : $$1+\frac{2}{6}+\frac{2\cdot 5}{6\cdot 12}+\frac{2\cdot 5\cdot 8}{6\cdot12\cdot 18}+\ldots$$  thought: I have tried to manipulate the series so that the sum upto n terms , can be simplified. But I can't figure out any profitable result. Is there a "win64" identifier in Qmake project files?  documentation does not mention other than unix / macx / win32.  So far I've tried using:  win32:message("using win32") win64:message("using win64") amd64:message("using amd64")   The result is always "using win32".  Must I use a separate project-file for x32 and x64 projects, so they would compile against correct libraries? Is there any other way to identify between 32-bit and 64-bit environments? Why is the even root of a number always positive? Let $n \in \mathbb N$ be a natural number and $a \in \mathbb R$ be a real number. The $n$-th root of the number $a$ is defined as follows:  Case I: $n$ is an odd number. In this case the $n^{\text{th}}$ root of $a$ is defined to be that number $b \in \mathbb R$ such that $b^n = a$.  Case II: $n$ is an even number. In this case the $n^{\text{th}}$ root of $a$ is defined to be that number $b \geq 0$ such that $b^n = a$.  Why is it that when $n$ is even, we only consider $b \geq 0$. For example, both $+2$ and $-2$ squared equal $4$, but when we say the square root of $4$ is $2$. Is there a reason for this? How many rectangles or triangles. For what values of $n > 1$ and a is $x^n - a$ irreducible in $\mathbb Q[x]$? $$x^n-a$$  So $n$ is any integer greater than 1, and $a$ is any integer.  $a$ being any integer is where I am running into trouble.  I have already shown and worked out a proof for this being irreducible when a is prime.  Now I am just working through examples, trying to figure out a pattern.  The exercise is asking for application of Eisenstein's Criterion only, so I am assuming that the value of n should have no effect on the irreducibility. What does really attracts a water stream to a charged object? I saw  by  and I was not 100% sure about the third phenomena, a statically charged object attracting a stream of water, especially because he explicitly mentioned that it is not due to the  of the water. A week later he uploaded an  all five phenomena and he says it is because of repelled equally charged ions, back up the water flow, such that the water actually falling along the charged object would be oppositely charged and therefore attracted. Here is a still image from the video illustrating this:    However this explanation seems strange to me, since this would mean that this way you could build up a charge in the water which has exited the tap (as well as in the water inside the tap as well). Therefore it should get harder to push those ions back over time, and therefore there also would be a decrease in the attraction of the water (assuming there is no other way to discharge it). I have tried a simple similar experiment, where I catched the water leaving the tap in an insulated bowl, but I did not notice a decrease in attraction.  There a couple of reasons why my experiment does not rule out his theory:   I was holding the charged (plastic) object by hand, which caused that its distance to the water stream varied, making it difficult to observe a decrease in attraction. I only conducted my experiment for roughly 30 seconds, which might be enough time for a opposite charge to build up.   The reason that he mentioned why permittivity can't cause this phenomena is because the electric field would be homogeneous and therefore the attractive and repulsive force of both dipoles of a water molecule would cancel out. However the electric field would only be homogeneous if the charged object would be an infinitely large plate, which it isn't. Thus water molecules would be attracted to the charged object. In order to make the permittivity theory more plausible I thought of maybe doing a order of magnitude calculation to see if it could generate a large enough force would be able the water stream by the observed amount. However I do not know what a good guess would be for variables such as the amount charge on the charged object and how the electric field would fall off over distance.   So, my question is whether his theory is really correct, or if the permittivity of the water can cause it. Easy way to determine the virtualization technology of a Linux machine? Sending a value to JavaScript I have set theme handler for a form in Drupal and this handler has a template file. What is the easiest way to send a value from PHP to JavaScript in that include file? jQuery Event Keypress: Which key was pressed? With jQuery, how do I find out which key was pressed when I bind to the keypress event?  $('#searchbox input').bind('keypress', function(e) {});   I want to trigger a submit when ENTER is pressed.  [Update]  Even though I found the (or better: one) answer myself, there seems to be some room for variation ;)  Is there a difference between keyCode and which - especially if I'm just looking for ENTER, which will never be a unicode key?  Do some browsers provide one property and others provide the other one? So I did a fresh install of Ubuntu 20.04 on a computer with the new RTL8125B Realtek network card and I cannot get the network to work. I was very suprised since this is supposed to be the new feature of the 5.4 kernel (I have 5.4.0-26-generic installed). Does anyone have any idea on how to get it working? Thank you. $ lspci -knn | grep Eth -A3 06:00.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd. RTL8125 2.5GbE Controller [10ec:8125] (rev 04) Subsystem: ASUSTeK Computer Inc. RTL8125 2.5GbE Controller [1043:87d7] Kernel modules: r8169 07:00.0 VGA compatible controller [0300]: Advanced Micro Devices, Inc. [AMD/ATI] Ellesmere [Radeon RX 470/480/570/570X/580/580X/590] [1002:67df] (rev e7)  $ dmesg | grep -i r8169 [    0.911236] r8169 0000:06:00.0: unknown chip XID 641 Coffeescript existential operator sometimes checks against undefined, but not always $\forall x,y>0, x^x+y^y \geq x^y + y^x$ Prove that $\forall x,y&gt;0, x^x+y^y \geq x^y + y^x$   A friend of mine told me none of the teachers in my school have succeeded in proving this seemingly simple inequality (it was asked at an oral exam last year). I tried it myself, but I made no significant progress. Hide some fields in the NewForm.aspx for a custom list I have 5 columns in the custom list. Out of which only 3 have to displayed in the NewForm.aspx page.  All of the 5 columns have to be displayed on the EditForm.aspx page. While googling, I have come across the following  :  When I implemented the code that was given on this page, I get an error which says that parentNode is not found. Please guide me how can I implement this requirement.  Any other alternatives other than this are also appreciated. Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction Given function $g$ continuous in line $[0,\infty)$, positive and strongly increasing that upholds $\lim_{t\to\infty}  g(t) = L$. Calculate the limit:  $$ \lim_{n\to\infty} \int\limits_1^2 g(nx) \, dx. $$  At the moment what I think that I should do is swap the $g(nx)$ with $L$ as it approaches infinity but from there on I am just not sure whats my next step is. A better proof for $\det(P) = \pm1$ if $P$ is an orthogonal matrix Set::setraw error in implementation of bisection method Make your own .sty files Sorry to start a subjective question, but want to understand "own.sty" file.  Other than all \usepackage{} what sort of other commands can be added to the makeyourown.sty file.  Can you include \let command, and \newcommands in it.   What sort of precaution / structure newbies should follow. How to copy files that have too long of a filepath in Windows?