Prove that two mutually exclusive events with positive probabilities can’t be independent.
What is the difference between independent and mutually exclusive events?
How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$? I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it by methods of complex analysis. But I have not taken this course yet. Thanks for any help. Sharing a Bluetooth Keyboard and Mouse sequentially between two Macs I work out of my house where I have two computers: a mid-2010 Macbook Pro (for my day job) and a mid-2010 iMac (personal). During the workday, my iMac acts as a second display for my Macbook Pro via Mini Displayport.  Currently, I use a USB keyboard and mouse for my work computer and a Bluetooth keyboard and Magic Mouse + Magic Trackpad for use with my personal computer. I'd like to get rid of the wired keyboard and mouse and just use the Bluetooth stuff for both machines (but never simultaneously).   Is it possible to easily share the single set of input devices without having to repeatedly break and reestablish the pairing of the input device to the correct computer? I'm not opposed to buying some type of additional hardware to accomplish this, provided it's not prohibitively costly or cumbersome to use or configure.  It's not clear to me how I would tell the MacBook Pro to leave the devices alone so the iMac can have them for the evening. I want to avoid de-pairing and re-pairing the devices each time I want to shift them back or forth.  Basically is there software or hardware that works as a Bluetooth KVM switch? Many companies offer index funds and ETFs. If two funds or ETFs track the same fund, what differentiates the two? Is it common for one index fund to outperform another? How can this happen if the funds/ETFs are tracking the same index?  I found . When forced to choose between them, index ETFs look like the better option (under the right circumstances), but what differentiates index funds/ETFs when compared to similar vehicles? Pay real money to set bounties (another method, not replacing rep method) It may seem like a stupid thing, but could we please be able to pay real money to set a bounty.  For example, if I had only 1 rep, I want to be able have a bounty.   Here late the suggested prices in USD. Convert money    $0.49 for each 50 reputation $0.99 for each week $1.99 for a month $9.99 for a year   Every month, the bounty fee would come in. This would mean you would have to enable your account with a bounty fee button on your profile. I want players to be able to click text to run an OP command even if that player is not an OP. For example something like:  Console: Do you want to enable Fly? [If you do, Please click here]  *Player1 clicks that text*  Console: /fly Player1   Is something like this possible? Irreducible polynomials and affine variety Let $k$ be any field, and let $f,g\in k[x,y]$ be two irreducible polynomials such that $g$ is not divisible by $f$. Prove that $V(f,g)\subseteq A_k^2$ is finite. Maintenance plan for creating transaction log backups I have a database that is set to the 'Full' recovery model, and I'm creating a maintenance plan to backup the Transaction Log every 15 minutes. A full backup is then taken once every 24 hours.  I was wondering: will the maintenance plan for the transaction logs create a lot of different files every 15 minutes, or will it by default append to the same file for 24 hours?  Also: what is preferred? I couldn't find anything on this subject (or maybe I was Googling the wrong words).  Thanks for any pointers anyone can give me.  EDIT: I also see that the checkmark "Verify backup integrity" is OFF by default. Is there a risk with turning it on? It seems to be that it should be on by default, so is there a reason for not using this? Cartographic techniques for symbolizing routing data Some common problems with displaying routes created from street data, such as bus routes, include:   Routes may overlap themselves, such as when a route takes the same road in opposite directions Routes may overlap other routes that take the same street. Routes are often overly detailed, or "noisy", owing to their derivation from street data. Indicating a route's direction unambiguously without overloading it with arrows can be a challenge.   What are some techniques for dealing with these issues in order to create a practical and usable route map, such as this ? Do you know of any such public transport maps that were produced by automatic techniques rather than manual illustration/digitization?  Specifically I am using ArcGIS 10, but solutions involving other software are welcome as well.  One technique I have had some success with for dealing with 1) is using ArcMap's  with a negative offset -- assuming the lines are oriented in the direction of travel, this will offset the lines such that the right and left-hand sides are distinct and oriented as you would expect (at least for  countries, for left-hand traffic countries one would use a positive offset). However the result tends to be a bit wonky around turns, interchanges and and U-turns, and is inflexible when dealing with multiple routes.  Another possibility that I am not very familiar with but which seems promising is  -- however, I have not had much luck in finding examples of using representations for transportation maps and whether it might fit the bill. How do I stop friends from getting notifications about what I do on Facebook? Friends and family (who are not in my Close Friends group) are getting notifications when I post or like something on Facebook. How do I stop this? To env or not to env Possible Duplicate:             Each post has an indicator which summarises who the post is shared.    What is the difference between these two terms? Should I write summarises or summarizes? HDD SMART interpretation - Possible impending drive failure I've just had an issue pop up related to my drives.  First things first, I was working in my case when I clumsily shifted the foot of the case off of my desk. Case shifted and hit the desk somewhat hard, but only fell an inch or two. It had been properly shut down so this may not have anything to do with the issue.  After booting up, I noticed that when playing a video from my F: Drive (Seagate), it produced a whine, almost a chirping sound coming from the hard drive bay. I think I may have heard this before the drop, but quieter.  Fearing impending failure I downloaded GSmartControl and inspected the drive information. The Seagate drive displayed a high Raw Read Error Rate, as well as a high Seek Error Rate.  I've read some of these values can be high on Seagate drives from the start. This drive is about 6 months old and the producer of the noise.  I also checked my Toshiba drive, which came with a refurbished prebuilt I bought about a year ago. It showed more normal SMART data, but displayed some concerning statistics (2 "Mechanical Start Failures" and 2 "Reported Uncorrectable Errors").  Both of the drives passed the basic SMART test, but the noise was quite alarming. It would stop as soon as a paused the video and start right up again as soon as I resumed playing.  Here is an album with 3 photos of the SMART data, the Seagate did not display statistics.  Could I please have help interpreting the SMART data, and whether this may indicate an impending failure?   Thank you for your time.   Here are images of the SMART data. What methods of suicide cause the least amounts of collateral damage? CK2 is a big game with lots of variables, so I'll stick to adding them upon being requested to minimize the hassle.  Anyway, let's say that you have a lot of children and the oldest one claims all the land according to laws of succession. After the father dies, the circumstances will obviously piss off a lot of people (in my case that would be 6 out of 8 vassals and 4 siblings with an opinion of -50 and below one of which is a sister married to the Basileus of the Byzantine friggin' Empire, a case of excommunication, 4 failed assassination attempts under 3 years... and lots and lots of vikings). Let's also say that the current playable character is completely incompetent compare to the next sibling in line.  So, what would be the most optimal methods to off myself without further screwing over the countryside so the next heir in line has to fix it (if we assume that said heir is actually capable of doing so)?  This is my list of DLCs (not including music and unit packs):   Charlemange Legacy of Rome Sword of Islam The Republic Way of Life Validity of work-energy theorem in presence of non-conservative forces? Spatial query to return multiple values I have a woodland scheme advice layer that has a polygon for each area of advice (brown layer in the picture). I also have a polygon layer of UK local authorities (red boundary layer in the picture). I would like to perform a spatial query which returns the local authority name for each woodland advice scheme. This would work fine if each woodland advice scheme was found in just one local authority boundary. However many are found across a boundary of two different authorities. For instance, one scheme sits across the boundaries of the authorities of Fife and Angus. The scheme is the yellow selected polygon in the picture and the red line is the authority boundary. How do I get the spatial join to return a value of Fife; Angus (or something similar to identify two values) instead of just Fife or Angus. Is there a term for solving a sub-sub problem of a problem so far removed from the main problem the link is not clear? In some situations, trying to solve a problem uncovers another problem. Attempts to fix that uncover yet another. A few more rounds and one can be doing something seemingly completely unrelated to the main problem. Is there an expression for that? Does the penalty to movement speed while grappling stack if you grapple two opponents? In 5e, a player who is grappling has their movement speed halved. However, a player with two hands can grapple two opponents.  As the wording seems to indicate that the movement speed penalty applies due to the grappling status, does that mean that a player grappling two opponents is doubly penalized?  If so, how would it apply: is your speed reduced to 0 ([1 - 0.5] - 0.5), or a quarter of your regular speed ([1 / 2] / 2)? When and how did "momentarily" come to mean "in a moment", rather than "for a moment"? "Momentarily" used to mean "for a moment" only, and not "in a moment". Thus, newscasters could be divided into two clear groups: those who would say "we'll be back momentarily," and those who would not.  This restriction made sense to me, because having both definitions would promote ambiguity if a unique interpretation could not always be derived from the context. But in recent years it seems "momentarily" is regularly, maybe even more often, used to mean "in a moment" by newscasters of every caliber, and in fact this is even shown to be the definition when looked up in most dictionaries.  When did this word's meaning change? How did it come about? Number of $(0,1)$ $m\times n$ matrices with no empty rows or columns Many to Many and Weak Entities I have an entity that cannot exist without being defined by another, and I want this entity to participate in a many-to-many relationship.  Example: An artist has an album (the album cannot exist without an artist), the album also has many tracks, but the same track can exist in many albums.  So we have a many-to-many relationship between the album and the tracks.  If the album is a weak entity, then its primary key is a foreign key referencing the artist, thus it cannot be a foreign key to another table representing a many-to-many relationship.  The question is: is it possible to have this kind of relationship in SQL, and if so, how do I express it?