How do I calculate the ratio of a distribution of random variables
Expectation of random variables ratio
What is the difference between ++i and i++? UI annoyances in the suggested edit review on Stack Overflow An inequality for a sum of sine functions I want to show for $x\in\mathbb{R}$, the following inequality holds. $$ \left|\sin x\right|+\left|\sin (x+1)\right|+\left|\sin (x+2)\right|&gt;8/5. $$ This trigonometric inequality has been verified by Mathematica using the Plot commend. However, using the sum formula for the sine function seems doesn't work, and I cannot give a rigorous proof of it. Any suggestion, idea, or comment is welcome, thanks! Find the value of : $\lim_{x \to \infty} \sqrt{4x^2 + 4} - (2x + 2)$ Remove trash icon from GNOME desktop and add to Ubuntu dock I just installed Ubuntu 18.04 and don't like having the trash icon on the desktop rather that in the launcher (Ubuntu dock).   How can I move it back to the launcher? Can other icons be moved from the desktop to the launcher? How to diagnose computer lockups and freezes? I built a desktop computer a couple years back with the following specs:  CPU: Intel  Yorkfield 2.5GHz 6 MB L2 Cache LGA 775 95W Quad-Core Processor BX80580Q9300 Motherboard:  LGA 775 NVIDIA nForce 680i SLI ATX Intel Motherboard Video Card: Two  GeForce 8600GT SCC 256 MB 128-bit GDDR3 PCI Express x16 SLI Supported Video Card PSU: SeaSonic S12 Energy Plus  ATX12V V2.3 / EPS12V V2.91 SLI Certified CrossFire Ready 80 PLUS Certified Active PFC Power Supply Memory: Two G.SKILL 4 GB (2 x 2 GB) 240-Pin DDR2 SDRAM DDR2 800 (PC2 6400) Dual Channel Kit Desktop Memory   Since its inception, the machine has periodically locked up, the regularity having varied over the years from once a day to once a month. Typically, lockups happen once every few days. By &quot;lockup&quot; I mean my computer just freezes. The screen locks up, I can't move the mouse. Hitting keys on my keyboard that normally turn LEDs on or off on the keyboard (such as Caps Lock) no longer turn the LEDs on or off. If there was music playing at the time of the lockup, noise keeps coming out of the speakers, but it's just the current frequency/note that plays indefinitely. There is no BSOD. When such a lockup occurs I have to do a hard reboot by either turning off the computer or hitting the reset button. I have the most recent version of the NVIDIA hardware drivers, and update them semi-regularly, but that hasn't seemed to help. I am currently using Windows 7 x64, but was previously using Windows Server 2003 x64 and having the same lockup issues. My guess is that it's somehow video driver or motherboard related, but I don't know how to go about diagnosing this problem to narrow down which of the two is the culprit.  Additional information re: cooling Regarding cooling... I've not installed any after-market cooling systems aside from two regular fans I scavenged from an older computer. The fan atop the CPU is the one that shipped with it. One of the two scavenged fans I added it located at the bottom tower of the corner, in an attempt to create some airflow from front to back. The second fan is pointed directly at the two video cards.  SpeedFan installation and readings Per studiohack's suggestion, I installed , which provided the following temperature readings:  GPU: 63C GPU: 65C System: 76C CPU: 64C AUX: 36C Core 0: 78C Core 1: 76C Core 2: 79C Core 3: 79C   Update #3: Another Lockup :-( Well, I had another lockup last night. :-( SpeedFan reported the CPU temp at 38 C when it happened, and there was no spike in temperature leading up to the freeze. One thing I notice is that the freeze seems more likely to happen if I am watching a video. In fact, of the last 5 freezes over the past month, 4 of them have been while watching a video on Flickr. Not necessarily the same video, but a video nevertheless. I don't know if this is just coincidence or if it means anything. (As an aside, each night before bedtime my 2 year old daughter sits on my lap and watches some home videos on Flickr and, in the last month, has learned the phrase, &quot;Uh oh, computer broke.&quot;)  Update #4: MemTest86 and 3DMark06 Test Results: Per suggestions in the comments, I ran the MemTest86 overnight and it cycled through the 8 GB of memory 5 times without error. I also ran the 3DMark06 test without a problem (see my scores at ). So... what now? :-) Any further suggestions on what to check? Is there some way to get a stack trace or something when the computer locks like that? Resolution I have never did figure out the particular problems, but based on the suggestions here and elsewhere, I'm presuming it was a motherboard issue. In any event, I recently upgraded my system, buying a new motherbeard, PSU, CPU, and RAM, and that new rig has been working splendidly the past several weeks. I am using the same graphic cards as in the old setup, so I think it's safe to reason that they weren't the cause of the problem. G-test vs Pearson's chi-squared test When would you use a WeakHashMap or a WeakReference? The use of weak references is something that I've never seen an implementation of so I'm trying to figure out what the use case for them is and how the implementation would work. When have you needed to use a WeakHashMap or WeakReference and how was it used? Can I update an early Airport Time Capsule? Finding roots/zeros for collision detection? For the most simple of 2D games, I have implemented a posteriori collision detection (overlapping rectangles) on the x/y Cartesian plane, but am now interested in understanding the basics of a priori collision detection...  In the link , I noticed the reference to Newton’s Method.  So, I want to better understand the link b/w finding zeros of an equation and detecting a posteriori collision.    Let's start with the most obvious case of finding roots of an equation:  If you have a projectile's trajectory modeled as a quadratic function, you can find the roots to predict at what time the height will equal zero (or any other constant height, actually, by just solving f(x)=C to be f(x)-C=0 and then finding roots of that expression).    So, as I understand, finding the root can tell me the time at which the object will be at any chosen height.  More broadly, we basically know the (x,y) location of the object for any given time.  But how does root finding, in particular, translate to detecting collisions with another object?  (The other object may be stationary, or be moving.  If the latter, do you also determine roots/location of this moving object as well? Calculate the (x,y) trajectory of both objects and determine where they will intersect?)  Again, the basic question is where does root finding come into play?  Any links to a basic primer on this topic, with a simple example would also be greatly appreciated. What happens with No-Maj-Borns? In the Harry Potter series, lots of Muggle parents are introduced to the wizarding world because of their magical children. This is seen with Hermione's parents visiting Diagon Alley and various other moments.   In the 1920s USA world that has been created, rules about fraternising with No-Majs are very strict. Is there any evidence in any of the Pottermore articles or recently published film tie-ins which explain what happens with magical children born to No-Majs around this time? (Clearly certain events in the film suggest that they may have no procedure and that there can be side-effects of this - but I feel that those events must be rare!) Wikipedia entry on likelihood seems ambiguous I have  experience where empty spaces cause unwanted effects. It is not easy to trace the cause of these unwanted effects. In order to eliminate any doubt, I often overuse % as follows.   \usepackage% [% left=3cm, right=3cm% ]% {geometry}   or  \newcommand{\mycommand}% {% This is my command.% }   I got an extreme example that will break what Leo Liu said     In fact, only spaces which would be   output have to be removed by comment.   The following code does not produce output, but we cannot remove the % and leave a blank line between two elements of a list below.  \newpsstyle{gridstyle} { gridwidth=0.4pt, % griddots=0 }   Shortly speaking, where are the necessary places to be appended with % to remove unwanted spaces? What do the slashes mean in the character tables? I've been reading the PF core rulebook, but confused over some of the data in the Character tables, and not found anywhere in the book that explains it, so wondering if someone could explain it?  Taking the Monk class table for example:     Level | Base Attack Bonus | ..... | Flurry of Blows Bonus      1st       |       +0          |   ....    |        -1/-1      2nd       |       +1          |   ....    |        +0/+0      ...      ...      16th  |       +12/+7/+2   |   ....    |        +14/+14/+9/+9/+4/+4/-1      17th  |       +12/+7/+2   |   ....    |        +15/+15/+10/+10/+5/+5/+0      18th  |       +13/+8/+3   |   ....    |        +16/+16/+11/+10/+6/+6/+1   What do the slashes between the bonuses mean? Audits bug in the filtered review queue Just recently the . I've got  this morning which seems to be out of scope for my filtered list. I have filtered my list to only review VBA and Excel-VBA tags while the audit is tagged with perl http pipes which have nothing in common with VBA. I see a problem with the way the system currently works  Since I have a filtered list of tags, once I see a differently tagged question I already know it's an audit. Come on, that's too easy!  Can this please be fixed? update I have ran into  which I decided to edit - the question was tagged VBA and I thought it had nothing to do with VBA so I selected Edit which was the correct action. Now, once the first audit was clearly one to close - nothing to do with VBA and really easy to know that it actually was an audit - the second one was a very good audit. I am just thinking now if this is a proper bug or just a matter of malfunction in the system. I think the first one may have been made an audit for the wrong reason. Let's turn this into a discussion as I can't clearly work the system out. Regression with only categorical variables Does metric signature affect the stress energy tensor? It looks like this issue has , but it just happened again on Travel.SE. Perhaps this is another corner case? These two questions are closed as duplicates of each other.           I had the last vote on each, although in both cases, my vote was for "Unclear what you're asking", but the weight of votes in both cases were for the duplicate status. So if the existing checks are only in place at the time someone casts a close vote, that would explain how this one got in under the radar. Perhaps another check needs to be done when the question is actually closed, not just when close votes are cast?  And as @gnat points out, low-rep/logged-out users get a Redirect loop when trying to load either question.  NOTE: A moderator has already fixed this particular circular duplicate situation by closing one of the questions with a different close reason. Subfields of finite fields Ethernet indicator in OSX menu bar I am using OSX 10.7  When I move between different ethernet networks (e.g., at work using ethernet on an external display, at home using thunderbolt ethernet), the ethernet does not always connect. However, I don't get immediate feedback. Instead, I connect to wi-fi. However, wi-fi is both slower and less reliable than ethernet. Thus, I'd like to know whether I am connected to ethernet or not. Of course I can go into preferences and network settings to see whether I'm connected to ethernet, but it would be better if there was some form of indicator in the status bar, just as there is for wi-fi. Thus, my question:  Is there a way of getting an indicator of ethernet connectivity in the OSX menu-bar?  Perhaps there is a free application that can do this.