Minimum possible value of the probability $P(X=0)$ where X is a discrete non negative random variable
Given $E[Y]=1$ , $E[Y^2]=2$ and $E[Y^3]=5$, Y non negative integer random variable, find min of $P[Y=0]$
Suppose $ \alpha, \beta>0 $. Compute: $ \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx $ Suppose $ \alpha, \beta&gt;0 $. Compute:   $$ \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx $$   Here is what I do: $$\begin{align} \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx &amp;= \int_{0}^{\infty}dx\int_{\alpha}^{\beta}\sin (yx)dy\\ &amp;=\int_{\alpha}^{\beta}dy\int_{0}^{\infty}\sin(yx)dx\\ &amp; \\ &amp;   \qquad\text{let $ yx=u $}\\ &amp; \\ &amp;=\int_{\alpha}^{\beta}\frac{1}{y}dy\int_0^{\infty}\sin u du\\ &amp;=\int_{\alpha}^{\beta}\frac{1}{y}dy\left( -\cos u|_{\infty}+\cos u|_0 \right)\\ &amp;=\log\frac{\beta}{\alpha}(-\cos(\infty)+1)\\ &amp;=\log\frac{\beta}{\alpha}-\cos(\infty)\log\frac{\beta}{\alpha} \end{align}$$  But $ \cos(\infty) $ does not exist right? Does it mean the integral actually diverse?  Edit: The question comes from   Who can point out my mistake in the above deduction? Inform users of certain tags on the existence of a new SE-2.0 site and offer to automatically post the question there instead Take for example  both at SO and SU. Now that there's , (new) users asking a LaTeX question should at least be informed that there is a new Stack Exchange site dedicated to TeX and thus increases their chance of an answer. This should happen before the question is actually posted and ideally offer the option to post the question directly there instead (without the need to copy-paste).  LaTeX is just an example, there's also , ,  etc., and it is obvious that this can only be decided on a case-by-case basis which is why I certainly do not suggest migrating questions automatically when these tags are used. But informing users of that possibility would probably help the SE network in general. Not enough space in /var/cache/apt/archives/ Is it safe to cancel CHKDSK when it has been started without any parameters? I know that it is not a good idea to forcibly stop a chkdsk process that has been executed with parameters like /f and /r.   But is it safe to stop the process when it has been started simply by the chkdskcommand? Chkdsk prints out a lot of errors and is continuing very slowly.  I suppose it to should be safe to cancel the process since there should be only in read mode, correct? Integrating a function of tan inverse How would I carry out the following integration? $$\int_0^12\arctan x^2$$  I tried to substitute $x^2 = \tan\theta$ From there I wrote the integral as: $$\int_0^{\pi/4}\theta\cdot \frac{\sec^2\theta}{\sqrt{\tan\theta}}\ d\theta$$  Now, is my only option to use integration by parts? Or is there a better method to solve this integral? How do I get the SQLSRV extension to work with PHP, since MSSQL is deprecated? How to create a color of the same brightness and saturation? Let's consider three colors: Red, Green and Blue. For blue I created a specific shade, something like #F2F5F7. If you see this color, it has a certain level of shade, brightness and saturation value of blue.  I want to create the same equivalent color but for Green and Red, how to get that? How to demonstrate that there is no all-prime generating polynomial with rational cofficents? When I go to any github.com page in Chrome, I get a big ugly error:  You attempted to reach github.com, but the server presented a certificate issued by an entity that is not trusted by your computer's operating system. This may mean that the server has generated its own security credentials, which Chrome cannot rely on for identity information, or an attacker may be trying to intercept your communications. You cannot proceed because the website operator has requested heightened security for this domain.  The same thing happens (in Chrome and with curl) when I go to  too. This strange problem started about a week and a half ago. Here's what I see when I click the broken lock icon in the address bar:   But gist.github.com works just fine:   It doesn't work with curl either:  Everything works fine in Firefox. How can I fix my root CA problem? Here's what it looks like in Firefox:   Update: I noticed that the first certificate in the chain is different in my broken Chrome/Safari as compared to Chrome on my other computer.   (There's no nasty red X anymore because I trusted it in Safari.) See how the issuers are different? What can I make of that? Deciphering the RSA encrypted message from three different public keys Limit of a Piecewise Function defined on Rationals and Irrationals How can I reproduce the error "package... needs to be reinstalled, but I can't find an archive for it" for testing? From time to time, Ubuntu users encounter errors from  or , of the form:  The package some-package needs to be reinstalled, but I can't find an archive for it.  I want to know what causes this error so that I can reproduce it and try to find safe ways of fixing it, even when the broken package is important and has many dependencies.  of the  might give a bit of a clue, but I don't know how packages get the status that causes the error.  How can I produce this error for testing on my (Ubuntu MATE 17.10) system?  This is prompted by  and my long-standing discomfort with closing questions about this error against , whose answer uses dpkg --force-all to remove the offending package.  suggests this solution has worked for many users, but I am not confident that using --force-all won't cause later problems, or that it's a good idea to use this method to remove important packages like APT. Add a transparent image on top of a material I am trying to mount my ext4 (sdc6) partition from a live usb because that's how I deleted two unused partitions left from a prior install of ubuntu, using GParted. The swap is mounted. However, I can't edit fstab, or find anywhere to click in nautilus to mount sdc6. In the terminal, I get the error: "can't find /dev/sdc6 in fstab". I'm afraid to exit the usb and update grub if that partition isn't mounted. Is there any easy way to do this, or do I have to install gksu or some other editor? Or do I need to mount it as sda6?  Device     Boot Start     End Sectors  Size Id Type /dev/sda1  *     2048 3905535 3903488  1.9G  b W95 FAT32  Device     Boot     Start        End   Sectors   Size Id Type /dev/sdc1            2048   29362175  29360128    14G 27 Hidden NTFS WinRE /dev/sdc2  *     29362176   29566975    204800   100M  7 HPFS/NTFS/exFAT /dev/sdc3        29566976  487023432 457456457 218.1G  7 HPFS/NTFS/exFAT /dev/sdc4       487024638 1250263039 763238402   364G  5 Extended /dev/sdc5       728393728  740290559  11896832   5.7G 82 Linux swap / Solaris /dev/sdc6       487024640  728391679 241367040 115.1G 83 Linux  Partition table entries are not in disk order. Is it possible to fail an ability check on purpose? Here are the general guidelines regarding ability checks:     The GM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure. When the outcome is uncertain, the dice determine the results.   In some cases, you could want to fail an ability check. If you are climbing a wall an want to drop off, you fail simply because you stop attempting to climb, but in other cases, success and failure are both tied to the actual action and may both have desirable outcomes.  Example 1: Spell scrolls  An evil adversary wants you to cast a spell scroll for him, but you have him figured out and want to destroy his spell scroll instead, seemingly by accident. The rules on spell scrolls read:     If the spell is on your class’s spell list but of a higher level than you can normally cast, you must make an ability check using your spellcasting ability to determine whether you cast it successfully. The DC equals 10 + the spell’s level. On a failed check, the spell disappears from the scroll with no other effect.   A failure means that you successfully destroy the spell scroll.  Example 2: Amulet of the Planes  You are in possesion of the Amulet of the Planes. Your party is being overrun by terrible monsters, and the only way to save your party is to sacrifice yourself by sending yourself and the monsters to another plane of existence.     While wearing this Amulet, you can use an action to name a location that you are familiar with on another plane of existence. Then make a DC 15 Intelligence check. On a successful check, you cast the Plane Shift spell. On a failure, you and each creature and object within 15 feet of you travel to a random destination. Roll a d100. On a 1-60, you travel to a random location on the plane you named. On a 61-100, you travel to a randomly determined plane of existence.   By walking up to the monsters and failing the check, you and the monsters would be sent to another plane of existence.  The question  The rules seem to assume that you always want to succeed when you perform an ability check. Sometimes, however, you could want to fail on purpose. A sensible DM would rule that if you are skilled enough to succeed the check, then you would undoubtly be able to fail it on purpose as well. What does the rules as written say -- if anything -- about this circumstance?  According to RAW, is it possible to attempt an ability check and intentionally fail? WinError 32 error when trying to delete shapefiles with QGIS python I always thought of current as the time derivative of charge, $\frac{dq}{dt}$. However, I found out recently that it is the ampere that is the base unit and not the coulomb. Why is this? It seems to me that charge can exist without current, but current cannot exist without charge. So the logical choice for a base unit would be the coulomb. Right? So what does Cyanogen Inc shutting down mean for CyanogenMod and similar ROMs? So Cyanogen Inc (cyngn), known for their support for their support for CyanogenMod and their efforts to integrate Cyanogen OS into devices such as the OnePlus One, Alcatel POP Mirage, and . Cyanogen Inc just recently .  What does this mean for CyanogenMod, Cyanogen OS, and the devices which originally shipped with Cyanogen OS? Will they no longer receive OTA updates? Checking to see if a DateTime variable has had a value assigned "He needn't say" / "He doesn't need to say" When I studied English at school (in Italy, more than 35 years ago), we were taught a rule according to which one should say, e. g., "He needn't (need not) say", but NOT "He doesn't need to say".  If I'm not mistaken, it was explained to us that the rule holds good with the following limitations:  1) only in the negative form: thus, "He needn't say", but "He needs to say", and "Does he need to say?";  2) only in the present tense: thus, "He needn't say", but "He didn't (won't, wouldn't, etc.) need to say";  3) only when an infinitive follows: thus, "He needn't see a doctor", but "He doesn't need a doctor".  But, if I google "doesn't need to", I find 125,000,000 occurrences.  I was wondering: does the rule hold good more for British than for American English?  Also, perhaps it isn't an absolute rule, and both constructions ("He needn't say" and "He doesn't need to say") are possible.  Is the rule at least a good description of the circumstances when we CAN (but needn't necessarily) use this construction?  Thank you very much, for this and the other answers.