Suppose that $X_1, X_2, \cdot \cdot \cdot \ , X_n$ is a random sample from a continuous distribution with pdf $f_X(x;\theta) = \theta x^{\theta-1}$, for $\ 0\leq x \leq 1$. Show that $W=\prod_{i=1}^n X_i$ is a sufficient statistic for $\theta.$ Is the maximum likelihood estimator for $\theta$ a function of $W$?  Here's what I did:  $$P(X_1=x_1,...,X_n=x_n|W=w)=\cfrac{P(X_1=x_1,...,X_n=x_n,\prod_{i=1}^nX_i=\prod_{i=1}^nx_i)}{P(\prod_{i=1}^nX_i=\prod_{i=1}^nx_i)} \stackrel{redundancy}= \cfrac{P(X_1=x_1,...,X_n=x_n)}{P(\prod_{i=1}^nX_i=\prod_{i=1}^nx_i)}=\cfrac{\theta^n(\prod_{i=1}^nx_i)^{\theta-1}}{P(\prod_{i=1}^nX_i=\prod_{i=1}^nx_i)}$$  Here I am stuck. I am having difficulty breaking up the probability in the denominator. Thanks for the help.
This problem comes from , who do not rigorously demonstrate (in their solution key) that the statistic is not sufficient.  Let $X_1,\dots,X_n$ be a random sample from a population with PDF $f(x|\theta)=\theta x^{\theta-1} \cdot 1_{\{x\in(0,1)\}}$ for $\theta&gt;0$. Show that $\sum_i X_i$ is not sufficient for $\theta$.  If you write out the PDF $p(\vec{x}|\theta)$ of the random sample, it is clear by the factorization theorem that $\prod_i X_i$ or $\sum_i \log(X_i)$ are sufficient for $\theta$; the PDF $p$ also suggests $\sum_i X_i$ is not sufficient for $\theta$. However, to rigorously demonstrate this, we need to analyze $p(\vec{x},\theta)/q(T(\vec{x},\theta)$, where $p$ is the distribution of sample $\vec{x}$, $q$ is the distribution of the statistic $T(\vec{X})=\sum_i X_i$. But finding the distribution of $T(X)=\sum_i X_i$ seems impractical. I have observed that $f$ is the PDF of a Beta($\theta$,1) distribution, but checking online, the distribution of the sum of Beta random variables doesn't appear to have a closed form. Are there any alternative routes (e.g., showing there is no factorization involving $T(\vec{X})$)? Did C&amp;B leave out a full explanation that $\sum_i X_i$ is not sufficient because none actually exist?
How to work with iAds using xcode 3.1.3 When does a flying creature start and end falling? Situation:  Alice the Aarakokra sees Bob the Battlemaster coming. She ascends to 605', just out of the 600' range of Bob's longbow.  Bob (Str 14) performs a running high jump 5' into the air and successfully hits Alice with his longbow, despite firing at disadvantage for long range. He spends one of his maneuver dice to perform the Trip Attack maneuver, and succeeds on the contested roll to knock Alice prone.  Alice is now flying and prone, so she falls. Falling does 1d6 bludgeoning damage per 10' fallen, to a maximum of 20d6.  Questions:   When does Alice start falling? It is on Bob's turn as soon as she's knocked prone, or some time else such as the beginning of her own next turn? When does Alice hit the ground? On Bob's turn as soon as she's knocked prone, or some time else?   These answers determine whether anyone can save Alice from her 20d6 of certain doom.   Is a reaction (e.g. casting Feather Fall) on Bob's turn the only hope? Can Alice's ally Charlie swoop in on his subsequent turn within the same round to catch her? Can Alice spend half her movement on her next turn to "stand from prone" and pull out of her involuntary dive? If Alice had expected to be shot down, could she have spent an action to be Ready to "stand from prone" while moving through Bob's airspace? Is it acceptable to change the vid column of a term programmatically in the database? I need to move some terms from one vocabulary to another by retaining all their data: translations, aliases, and node references.  Is it acceptable to change its vid column value directly in the database? The terms don't have hierarchy relationships which needs to be handled. Taking screenshot of a specific area from the command line Strange character when using listings package Why did the Nazgûl not attack a second time at Weathertop? Can anyone help me with this limit problem without L'Hopital rule and Taylor series? $$\lim_{x\rightarrow\ -\infty}\left(-2\frac{\arctan(x)}{\pi}\right)^x$$ Cannot delete directory with Directory.Delete(path, true) I'm using .NET 3.5, trying to recursively delete a directory using:  Directory.Delete(myPath, true);   My understanding is that this should throw if files are in use or there is a permissions problem, but otherwise it should delete the directory and all of its contents.  However, I occasionally get this:  System.IO.IOException: The directory is not empty.     at System.IO.__Error.WinIOError(Int32 errorCode, String maybeFullPath)     at System.IO.Directory.DeleteHelper(String fullPath, String userPath, Boolean recursive)     at System.IO.Directory.Delete(String fullPath, String userPath, Boolean recursive)     ...   I'm not surprised that the method sometimes throws, but I'm surprised to get this particular message when recursive is true.  (I know the directory is not empty.)  Is there a reason I'd see this instead of AccessViolationException? Let's say I have some data, and then I fit the data with a model (a non-linear regression). Then I calculate the R-squared ($R^2$).  When R-squared is negative, what does that mean? Does that mean my model is bad? I know the range of $R^2$ can be [-1,1]. When $R^2$ is 0, what does that mean as well? I installed Ubuntu 12.10 and entered a password in SeaHorse. I would like to remove the password of "Default" but I get this error :  GDBus.Error:org.freedesktop.Secret.Error.NoSuchObject: The collection does not exist   Is there a fix or a way to bypass modify the password ? Are Canon EOS and Nikon DSLR lenses interchangeable? Can we use the lenses from, say a Canon EOS 550D kit (Canon EF-S 18-55mm) with a Nikon DSLR D5000 for example) ? Also, is the reverse possible ?  I'm looking at interchangeability between recent DSLR lenses only, and not with older lenses. Also, for "average" lenses, and not ultra wide/ultra zoom.  The reason for this question is that I'm trying to decide between a Nikon D5000 and a Canon EOS 500D. I like the Nikon better, but Canon is more popular here amongst people I can interchange lenses with :) A better serial voting trap Currently we capture fanatic, voting ring or rage-type activities, by measuring the number of up-votes or down-votes between the same two users in a certain (relatively short) time period. When this is detected by the system, the votes are reversed, and I've never bothered to investigate what else happens after that - though I'm sure some form of wrist-slapping ensues.  While it is intended to be well-guarded for obvious reasons, the algorithm used is not exactly a secret, and I am sure there are plenty of examples where people are dangling (maybe intentionally, or maybe by dumb luck) just on the safe side of getting caught.  Personally I believe there are active, intentional, long-term vendettas in operation right now where users either:   blindly up-vote every answer from a user; or, down-vote a particular user once or twice a day, regardless of topic or quality.   Now, maybe the first one is valid - if I'm operating in the same tag-sphere as some smart person, it may just be that I'm constantly enthralled with their answers, and I'm not even noticing who they're written by. (Though, even Skeet isn't perfect, and I haven't up-voted every answer I've seen him post.)  But still, can we do a better job of catching - or at least noticing - this activity over the long term in addition to the short term?  The behavior is the same, and shouldn't be tolerated, regardless of the time frame - vengeful rage serial voting, blind staker crush serial voting, and painstakingly patient serial voting are all undesirable behaviors IMHO.  I'm not saying it has to do anything automatically, as I'm well aware of the possibility of false positives. And I'm not even sure what to propose as a time frame or a vote threshold.      But if a large percentage of my up-votes or down-votes are against a particular user, and especially if on a majority of those posts I am the only one voting in that direction, I think it should at least be raised as some kind of flag to moderators.   This would work much like how other suspicious activity is highlighted (but also not dealt with automatically). If a pattern becomes obvious and persists, then they can intervene in some way (or not, but at least they were provided with the information - just like they can reject and ignore manual flags). Today, all I can do is e-mail team@stackoverflow.com, and unless I know exactly which person(s) are involved, there is not much they can do about it.  And you're thinking, "Meh, let some hater down-vote you once or twice a day. Unless all you're posting is utter crap, it should be statistically insignificant." You're right. But reputation is not the point, and is not the motivation behind my request. What I'd like to see curtailed is the tendency for folks to down-vote a question or answer because of who posted it, not because of what it said or didn't say. When a down-vote is personal it can leave a person second-guessing the quality of their post. Which isn't necessarily a bad thing, except when it happens repeatedly.  Sorry for the essay but I wanted to try to anticipate and cover all the objections and hole-poking. I think there's an unintended motivation for users to post questions to higher-traffic Stack Exchange sites rather than their more specific or appropriate lower-traffic ones.  I think I have a solution.  What motivated this suggestion for me was that I am confident that if users were to post Android design questions to  beta and/or , that they would likely not be noticed by the appropriate audience compared to the vast army of Android experts here on Stack Overflow.    There are plenty of Android questions on SO that really should be on one or both of the aforementioned sites.  Presently there is little motivation for users to start questions like this on the appropriate site.  Is there something we could do to cross-pollinate related sites?  For instance, what if we let certain tags span certain sites?  Obviously the selection would be controlled by high-reputation users.   There could also be a feature to show (or explicitly hide) certain questions from other sites regardless of the tags.  This could be controlled by lower-reputation users than the global tag since it wouldn't have much statistical impact.  For instance, the android-layout tag could by default span sites, but OPs and other users could eliminate from the Graphic Design site questions that are of a purely technical nature.  Users interested in Android tagged questions would see related questions on design and UE sites--whether they are searching, posting, or viewing a question through the related items.  This would help eliminate cross-site duplicates, and encourage people to post on the correct site in the first place.   For related question lists and search results, there could even be a specific cross-site icon  next to an off-site question to show that it's from UX, for example. This is what I'm trying to do:  I have a website A where I make configuration changes and add other stuff. Now I create a website B. I would like to add the configuration changes from website A to this website.   I know the UUID of both websites have to be the same (like staging &amp; production environment). In "" they explain how to do this with drush cget system.site.   The problem is when I run drush cget system.site in my website A I get the following drush error:     Command config-get needs a higher bootstrap level to run - you will need to invoke drush from a more functional Drupal environment to run this command.                                                               [error]   The drush command 'cget system.site' could not be executed.                                                                                                                                                           [error]   Drush was not able to start (bootstrap) the Drupal database.                                                                                                                                                          [error]   Hint: This may occur when Drush is trying to:    * bootstrap a site that has not been installed or does not have a configured database. In this case you can select another site with a working database setup by specifying the URI to use with the --uri parameter   on the command line. See drush topic docs-aliases for details.    * connect the database through a socket. The socket file may be wrong or the php-cli may have no access to it in a jailed shell. See  for details.   Drush was attempting to connect to:  Drupal version         :  8.0.0-rc1  Site URI               :  http://default  Database driver        :  mysql  Database hostname      :  localhost  Database port          :  3306  Database username      :  root  Database name          :  s_1  PHP executable         :  /Applications/MAMP/bin/php/php5.5.10/bin//php  PHP configuration      :  /Applications/MAMP/bin/php/php5.5.10/conf/php.ini  PHP OS                 :  Darwin  Drush script           :  /Users/myname/.composer/vendor/drush/drush/drush.php  Drush version          :  8.0-dev  Drush temp directory   :  /tmp  Drush configuration    :  Drush alias files      :  Drupal root            :  /Applications/MAMP/htdocs/s1  Site path              :  sites/default  Sync config path       :  sites/default/files/config_bZM1aJhLcWGpFZJEzfEw_pKF4g71jOdkSRti30k540BAwE9GQ4U6p469a9VxyJS4j4JVfC58qg/sync   I've installed the latest version of drush .. . I'm working with the release candidate of Drupal 8. Is there another way to get my UUID? Or do you have a tutorial or a video/blogpost of somebody who already did this?  UPDATE:  Ok, I was able to retrieve the UUID from website A and changed the UUID in website B so they have the equal UUID.  After that I changed the system site slogan of website A.  Then I did an export of the configuration in website A. But when I tried to import it in website B I got 137 new files to sync .. . The UUID is different in all files like this:    When I tried to synchronize I got the following error: I can never figure out whether I should use who and whom. Most people use who for both colloquially, but some people say this is not correct.  What’s the rule for using who and whom correctly? I was born in Brazil and came to the United States at the age of 7, in 2007. I no longer need to live in the United States because I have found success in investing and no longer depend on having a job. I am going to buy property in my home country soon, but I have just turned 18.   Will I ever be able to return to the United States to see my friends and other family members? Even if it is a 30-day visa? And, if so, in the future, am I still eligible for the EB-5 visa? Will I be barred from entry? What should be done when no more Pokemon can be caught? (when full storage) It seems that I can buy 50 more slots using 200 coins, but what if I keep on getting more and more common Pokemons such as Pidgey or Rattata, then the slots will be full again.  Right now, I am sorting by CP (Combat Power) and then "Transfer" the lowest CP Pokemons to the Professor (to give it away), but wonder what is a good or better way to handle this?  (For example, can I combine several Pidgey into one?) I just added a PPA repository for the development version of the GIMP, but I get this error:  $ apt-get update &amp;&amp; apt-get upgrade ... The following packages have been kept back:   gimp gimp-data libgegl-0.0-0 libgimp2.0   Why and how can I solve it so that I can use the latest version instead of the one I have now? Non-blocking Synchronous AJAX I told my wolves to sit down and I accidentally forgot to tell them to stand up and I left them. Now I cannot find them, how do you get back to your wolves if they are lost?