Suppose that $X_k$ are iid $\mathrm{Bernoulli}(1/2)$for $k=1, \ldots, n$ and  $$Y_n = \sum_{k=1}^n \frac{X_k}{2^k} $$ Given $U\sim \mathrm{Unif}(0,1)$ then the mgf of $U$ is $$m_U(-s) = \mathbb{E}(\exp(-sU)) = \frac{1-\exp(s)}{s} $$ (they gave it like this).    Show that $Y_n$ converges in distribution to $U$ by showing $$\lim_{n\to\infty}\mathbb{E}(-sY_n) = \frac{1-\exp(s)}{s}.$$    I computed the left hand side to be $$\mathbb{E}(-sY_n) = \frac{1}{2^n}\prod_{k=1}^n (1+\exp(-\frac{s}{2^k}))   $$ and am now stuck on how to take the limit. Also, how does showing the limit converging of the MGF imply they converge in distribution?    (I'd like to do it by finding the limit of the MGF).
I know that the finite sum of Bernoulli i.i.d. random variables is a binomial distribution, but what is the distribution of $$\lim_{n \to \infty}\sum_{k=1}^{n} \frac{x_k}{2^k}$$ where $x_k$ is a Bernoulli random variable with parameter $\frac12$?
Unable to Change Apple ID and password I have tried repeatedly to change my Apple ID and password without success. I have followed all of the directions trying both my emails and asked for new passwords----no luck whatsoever. I was going to make a filter for the &quot;&quot; by excluding some sites from it.  I noticed  which talks about &quot;negative filtering&quot; but does not explain how one can do it or if it is possible.  There does not seem to be a way to do this, though.  On the filter page, my only options are:  So how can I do &quot;All sites except [list]&quot;? It would also be nice if this could include similar functionality with the meta checkbox.  Something like &quot;All main &amp; meta sites except...&quot; Do you have advantage when multiple sources give advantage and only one gives disadvantage? As I understand it:   if you have Advantage from two different sources, you simply have Advantage (no stacking) if you have advantage from one source, and disadvantage from one source, they cancel out.     So what if you have advantage from two or more sources, and disadvantage from one source?  My assumption is that you simply have advantage, or put another way, you have Advantage on a roll when the # of Advantages > # of Disadvantages (and vice versa).  Is that correct? Prevent Use of the Back Button (in IE) How to store pipe ( | ) in a variable? The idea would be to use it as... a pipe in a command. For instance: say there's some kind of long path which has to be retyped again and again, followed by a pipe and a second program, i.e.  "directory1/directory2/direcotry3/file.dat | less -I "   I'd like that part to be stored in a variable, so it could be used like this:  r="directory1/directory2/direcotry3 \| less -I -p " $ cat path1/path2/$r &lt;searchterm&gt;   Instead, I get  cat: invalid option -- I Try `cat --help' for more information.   ... meaning the pipe clearly didn't work. Windows update giving error 80070002 and manual update not working how to update? I have been trying to update my pc from 20h1 to 20h2 but the windows update shows error message 80007002 and when I manually install the update using windows 10 update assistant it downloads completely and then even starts install and gets stuck at 87 percent on the screen where it says do not turn off. What to do? This is the log file. Matching Profile found: FindRollbackFailure - 3A43C9B5-05B3-4F7C-A955-88F991BB5A48 SetupDiag version: 1.6.0.0 System Information:     Machine Name =      Manufacturer =      Model =      HostOSArchitecture = 1033     FirmwareType = UEFI     BiosReleaseDate = 20160513000000.000000+000     BiosVendor = A12     BiosVersion = A12     HostOSVersion = 10.0.19041     HostOSBuildString = 19041.1.amd64fre.vb_release.191206-1406     TargetOSBuildString = 10.0.19041.621 (vb_release_svc_prod1.201014-1758)     HostOSLanguageId =      HostOSEdition = CoreSingleLanguage     RegisteredAV = Quick Heal Internet Security     FilterDrivers = FileInfo     UpgradeStartTime = 12/20/2020 2:54:38 PM     UpgradeEndTime = 12/20/2020 5:30:40 PM     UpgradeElapsedTime = 02:36:02     CV = Pi3IdU3SlEqpFx93     ReportId =    Error: SetupDiag reports rollback failure found. Last Phase = Pre OOBE Boot Last Operation = OOBE boot apply Error = 0xC1900101-0x4000D LogEntry:   Refer to &quot;https://docs.microsoft.com/en-us/windows/desktop/Debug/system-error-codes&quot; for error information.  Last Setup Phase: Phase Name: Pre OOBE Boot     Phase Started: 12/20/2020 4:37:50 PM     Phase Ended: 1/1/0001 12:00:00 AM     Phase Time Delta: 00:00:00     Completed Successfully? False   Last Setup Operation: Operation Name: OOBE boot apply     Operation Started: 12/20/2020 4:37:50 PM     Operation Ended: 1/1/0001 12:00:00 AM     Operation Time Delta: 0:00:00:00.0000000     Completed Successfully? False Peppers evolved chemicals like capsaicin to discourage animals from eating them, and that chemical itself can cause internal damage if consumed in large quantities, though this is usually only a concern for the hottest of peppers (which people do eat).   Still, why is it that humans seem to be the only animal that often "like" spicy food? How do I delete a 'locked' file? On Windows XP, when I try and delete a specific file it says:     Cannot delete blah.blah: Make sure the   disk is not full or write-protected   and that the file is not currently in   use.   How can I go around this warning and delete it anyway? How to create python bytes object from long hex string? What is wrong with using inline functions? While it would be very convenient to use inline functions at some situations,  Are there any drawbacks with inline functions?  Conclusion:  Apparently, There is nothing wrong with using inline functions.  But it is worth noting the following points!   Overuse of inlining can actually make programs slower. Depending on a function's size, inlining it can cause the code size to increase or decrease. Inlining a very small accessor function will usually decrease code size while inlining a very large function can dramatically increase code size. On modern processors smaller code usually runs faster due to better use of the instruction cache.  The speed benefits of inline functions tend to diminish as the function grows in size. At some point the overhead of the function call becomes small compared to the execution of the function body, and the benefit is lost  There are few situations where an inline function may not work:   For a function returning values; if a return statement exists. For a function not returning any values; if a loop, switch or goto statement exists.  If a function is recursive.   The __inline keyword causes a function to be inlined only if you specify the optimize option. If optimize is specified, whether or not __inline is honored depends on the setting of the inline optimizer option. By default, the inline option is in effect whenever the optimizer is run. If you specify optimize , you must also specify the noinline option if you want the __inline keyword to be ignored. How much longer will my 10 year old frame last? I'm developing a messenger application with encrypted chats.   In the first version of the app I've used PBKDF2 (10000 iterations, SHA1, random salt) to extend a short user password and generate keys to encrypt (AES256) and sign message (HMAC).  In the current version of the app I generate a random 130-bit key and use it instead of a user password. I think that using PBKDF2 with such a key has some overhead.   I see 2 possible ways to improve the current approach:   Keep using PBKDF2 and decrease iterations to 1 or 10. Don't use PBKDF2 and use HKDF instead.   What are the advantages and disadvantages of each approach? Are there any other options? I am trying to copy specific attribute table rows from one shapefile to a new shapefile that will contain only the specified rows in its attribute table, that is, if a field GDP_TOP10 = 1.   I am using this at the moment:  #Grab rows using Search Cursor with WHERE clause rows = arcpy.SearchCursor("C:/CGDP.shp", "GDP_TOP10 = 1") for row in rows:     arcpy.CopyFeatures_management("CGDP.shp", "New.shp") del row del rows   However, this just copies the entire shapefile, not the rows I want to select with the cursor. If I execute a print statement under my for loop instead, it works and prints only the relevant fields, but the copy to new shapefile command does not seem to copy specifically.   Is there a different ArcPy command to only copy rows that match a query? android how an EditText work as AutoComplete Where to ask "subjective and argumentative" questions on programming? Which is faster/best? SELECT * or SELECT column1, colum2, column3, etc "Neither", "none", "no one" + [of them] + verb-s In a comment on English Language &amp; Usage I read the following answer responding to a(n) asker's question: &quot;Neither sound natural.&quot; I have always had more than one doubt on the &quot;neither&quot; usage and I would like understand(ing) the &quot;secrets&quot; to use this word better or, at the best, correctly. So, let us consider the following examples:  1 Neither sound natural. 2 None sound natural. 3 No one sound natural. 4 Neither of them sound natural. 5 None of them sound natural. 6 No one of them sound natural. 7 Neither sounds natural. 8 None sounds natural. 9 No one sounds natural. 10 Neither of them sounds natural. 11 None of them sounds natural. 12 No one of them sounds natural.  Comparing the above sentences (one) with each other, can anybody explain which is grammatical and why? Show that a Cauchy sequence has a fast-Cauchy subsequence How do I tell a script to wait for a process to start accepting requests on a port? I need a command that will wait for a process to start accepting requests on a specific port.    Is there something in linux that does that?  while (checkAlive -host localhost -port 13000 == false)   do some waiting  ... What is currently the best way to get a favicon to display in all browsers that support Favicons?