Let $X$ have the pdf $f_X (x;\theta)=1/(2\theta)$ for $-\theta&lt;x&lt;\theta$, zero elsewhere where $\theta&gt;0$ Is the statistic $Y=|X|$ a sufficient statistic for $\theta$?    My book claims it is but I do not see it. A transformation argument shows that the distribution of $|X|$ is uniform with $f_X (x;\theta)=1/\theta$ for $0&lt;x&lt;\theta$. Thus the conditional probability  $$\frac{f(x_1;\theta)f(x_2;\theta)\ldots f(x_n;\theta)}{f_{|X|}(|X|;\theta)}$$  does depend on $\theta$ and therefore, $|X|$ cannot be a sufficient statistic for $\theta$.  Is there something that I am doing wrong here?
So, I know that $\max(-X_{(1)},X_{(n)})$ is a sufficient statistic for the parameter $\theta$. But can I also say that $(X_{(1)},X_{(n)})$ are jointly sufficient for the parameter $\theta$ ?  In other words, can a single parameter have jointly sufficient statistics?
I'm experimenting with VPNs, so I wanted to install pptpd.  john@desktop:~$ sudo apt-get install pptpd [sudo] password for john:  &lt;snip&gt; The following extra packages will be installed:   bcrelay The following NEW packages will be installed:   bcrelay pptpd 0 upgraded, 2 newly installed, 0 to remove and 11 not upgraded. Need to get 90.5 kB of archives. After this operation, 442 kB of additional disk space will be used. Do you want to continue [Y/n]? y   And everything runs fine until I get down to here:  WARNING: The following packages cannot be authenticated!   bcrelay pptpd Install these packages without verification [y/N]? n E: Some packages could not be authenticated   No authentication?  I'm no security expert, but I'm fairly sure that's a bad thing.  So I .  john@desktop:~$ apt-cache policy pptpd pptpd:   Installed: (none)   Candidate: 1.3.4-3ubuntu1   Version table:      1.3.4-3ubuntu1 0         500 http://us.archive.ubuntu.com/ubuntu/ natty/main i386 Packages john@desktop:~$ apt-cache policy bcrelay bcrelay:   Installed: (none)   Candidate: 1.3.4-3ubuntu1   Version table:      1.3.4-3ubuntu1 0         500 http://us.archive.ubuntu.com/ubuntu/ natty/main i386 Packages   They're coming from Ubuntu.com and not being properly authenticated?  What's going on here? What does different focal length offer me? English proverb for when a solution comes too late In Flemish we have a saying "Vijgen na pasen". Translated: "figs after Easter". It means a solution comes too late to be of any use.  What is the English equivalent for this?  Some googling gives me "Closing the barn door when the cow has bolted", but the explanation seems to point more towards the wrong solution for a problem. How do I recruit beta testers? 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After that I want to add this app to Ubuntu Software Center for free Is it possible?   My app is just one *.jar file, its helping to play card games.  I dont know Python language and saying the truth - dont want to learn it now.  What should I do to be able to submit my app? What are attention mechanisms exactly? Attention mechanisms have been used in various Deep Learning papers in the last few years. Ilya Sutskever, head of research at Open AI, has enthusiastically praised them:    Eugenio Culurciello at Purdue University has claimed that RNNs and LSTMs should be abandoned in favor of purely attention-based neural networks:    This seems an exaggeration, but it's undeniable that purely attention-based models have done quite well in sequence modeling tasks: we all know about the aptly named paper from Google,   However, what exactly are attention-based models? I've yet to find a clear explanation of such models. 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Equivalent of typedef in C# Is there a typedef equivalent in C#, or someway to get some sort of similar behaviour? I've done some googling, but everywhere I look seems to be negative. Currently I have a situation similar to the following:  class GenericClass&lt;T&gt;  {     public event EventHandler&lt;EventData&gt; MyEvent;     public class EventData : EventArgs { /* snip */ }     // ... snip }   Now, it doesn't take a rocket scientist to figure out that this can very quickly lead to a lot of typing (apologies for the horrible pun) when trying to implement a handler for that event. It'd end up being something like this:  GenericClass&lt;int&gt; gcInt = new GenericClass&lt;int&gt;; gcInt.MyEvent += new EventHandler&lt;GenericClass&lt;int&gt;.EventData&gt;(gcInt_MyEvent); // ...  private void gcInt_MyEvent(object sender, GenericClass&lt;int&gt;.EventData e) {     throw new NotImplementedException(); }   Except, in my case, I was already using a complex type, not just an int. It'd be nice if it were possible to simplify this a little...  Edit: ie. perhaps typedefing the EventHandler instead of needing to redefine it to get similar behaviour. Getting text from last marker to EOF in POSIX.2 If the universe is expanding, what is it expanding into? When the big bang happened where did it occur?  When the big bang happened how did it occur?  Where did the energy come from? Energy can not be created or destroyed does that mean, energy has existed before the universe was here? How come SG-1 couldn't get back to Earth? Proof relation between Levi-Civita symbol and Kronecker deltas in Group Theory In order to prove the following identity: $$\sum_{k}\epsilon_{ijk}\epsilon_{lmk}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$$ Instead of checking this by brute force, Landau writes thr product of Levi-Civita symbols as: $$\epsilon_{ijk}\epsilon_{lmn}=\det\left|    \begin{array}{cccc}       \delta_{il}   &amp; \delta_{im}  &amp; \delta_{in}   \\       \delta_{jl}   &amp; \delta_{jm}  &amp; \delta_{jn}   \\       \delta_{kl}   &amp; \delta_{km}  &amp; \delta_{kn}       \end{array} \right| $$  The proof that the equalty holds is quite straightforward if you consider what values the indices can take. But I've been told that there's a much more profound and elegant demonstration based on the representation of the symmetric group.  Does anybody know this approach based on group theory? How  can I delete a node after, for example, 2 weeks it has been created? What is a good book to learn number theory?