Proving minimal sufficiency using Lehmann-Scheffe approach when joint pdf is only an indicator function
Minimal sufficient statistic of $\operatorname{Uniform}(-\theta,\theta)$
Given a compact Hausdorff space $X$, if $a \in X$ is a $G_\delta$ singleton (i.e., $\{ a \}$ is a $G_\delta$ set), then there is a countable local base at $a$.   The point $a$ can be written as countable intersection of open sets let's call $U_{n}$.  How can I use the compactness? What is the next step for proof? Could you give me hint please? Two arrays in foreach loop When you pass a collection like list, array to another function in python, does it make a copy of it, or is it just a pointer? We will make the convention that exact categories be skeletally small.   Is this construction (used in a mathematical context) correct? There is something that strikes me as odd in that "be". Should it be "are"? I know I've seen a similar way of using "be" like that, without conjugating the verb, but I'm not completely sure when it should be used. I am really having trouble working a problem in my course. I will post it and then add what I have thought etc.  Let p be an odd prime, then using the Euclidian Algorithm, calculated the GCD of of $x^{p-1}-1$ and $x^2+1$ in $\mathbb{Z} / p\mathbb{Z}[x]$  Also, how does $p=1 (mod4)$ tell us that  $-1$ is a square in $\mathbb{Z} / p\mathbb{Z}$, infact that $-1$ is a square here if and only if $p=1(mod4)$.  My thoughts:   I know that $$x^{p}-x=(x)(x-1)(x-2)…(x-(p-1))$$ and so that in this sense our roots will be the congruence classes of $1,2,…p-1$  and also that $$x^{p-1}-1=(x-1)(x-2)…(x-(p-1))$$  But I get lost when trying to apply the EA to these polynomials,  like how can I do it when p is arbitrary?  Preferably id like to be able to understand/see how to do the first part first i.e. using division with residue and then move on to the second part. Any insight/hint or solution to this would be great,  Thank you all  Update:  When I try to do division with residue, I just get  $$x^{p-1}-1=(x^2+1)(x^{p-3}-x^{p-5}+x^{p-7}-x^{p-9}...)$$  for example,  $p=7$  $$(x^{6}-1)=(x^2+1)(x^4-x^2+1)$$  for $p=11$  $$(x^{10}-1)=(x^2+1)(x^{8}-x^{6}+x^{4}-x^{2}+1)$$  which for specific cases tells me that 1 is the gcd, but I dont know how to generalize that for the case with arbitrary prime p. Assume that $X$ and $Y$ are two random variables such that $Y=E[X|Y]$ almost surely and $X= E[Y|X]$ almost surely. Prove that $X=Y$ almost surely.   The hint I was given is to evaluate: $$E[X-Y;X&gt;a,Y\leq a] + E[X-Y;X\leq a,Y\leq a]$$  which I can write as: $$\int_A(X-Y)dP +\int_B(X-Y)dP$$ where $A=\{X&gt;a, Y\leq a\}$ and $B=\{X\leq a,Y\leq a\}$.  But I need some more hints. Max length for HTML How to restrict the maximum number of characters that can be entered into an HTML &lt;textarea&gt;? I'm looking for a cross-browser solution. How to create a Sierpinski triangle in LaTeX? I have been trying to recreate the following image in LaTeX    and for me personally it has to be done with pdflatex. Although answers using pst-fractals From Jake I got a good start which led to the following code  \documentclass{standalone} \usepackage{tikz} \def\level{5} \usetikzlibrary{lindenmayersystems}  \begin{document} \begin{tikzpicture}[l-system={step=5pt, order=\level, angle=120},rotate=180]     \pgfdeclarelindenmayersystem{Sierpinski triangle}{     \symbol{X}{\pgflsystemdrawforward}     \symbol{Y}{\pgflsystemdrawforward}     \rule{X -&gt; X-Y+X+Y-X}     \rule{Y -&gt; YY}     }     \draw [black] (3,2) l-system     [l-system={Sierpinski triangle, axiom=X, anchor=north east},fill=white]; \end{tikzpicture}  \end{document}   There are however two problems with the code.   I layer a black triangle behind the lindenmayersystem? How do I keep the size of the triangle consistent with different number of recursion levels. Can Dementors die? Based upon , J.K. Rowlings has apparently stated "You cannot destroy Dementors, though you can limit their numbers if you eradicate the conditions in which they multiply, ie, despair and degradation."  Does this mean that the number of Dementors is always either increasing or remaining the same?  If they cannot be destroyed, does that mean they are immortal?  Or do they die of "natural causes"? Updating entities in response to collisions - should this be in the collision-detection class or in the entity-updater class? Why is accuracy not the best measure for assessing classification models? A cyclic group is a group that is generated by a single element.  That means that there exists an element $g$, say, such that every other element of the group can be written as a power of $g$.  This element $g$ is the generator of the group.  Is that a correct explanation for what a cyclic group and a generator are? How can we find the generator of a cyclic group and how can we say how many generators should there be? How to get my actual data from ContentNote API in Salesforce using SOQL? I noticed in this thread the OP successfully managed to connect a 30" cinema display to the new Macbook Pro USB-C, with full resolution:    I have the official Apple dual DVI adapter, going to a displayport > USB-C adapter like yours, however the screen does not fire up.  I then tried a multi adapter instead of the single displayport > USB-C, which worked but only at 1280 resolution:    Would the official Apple Dual DVI adapter not work? How can I withdraw my paper from a fake conference? What bike+equipment for a long daily urban commute? In the (distant) past I commuted 6 km (more than half an hour) each way, on a cheap bike with big knobbly tyres.  Soon I'll have a new commute: 18 km each way, daily. That's much further than I've done in the past, and I don't want it to take 2 hours each way; but  suggests that it's feasible:      Most people who reside 10 to 20 kilometres away will also find it quite easy to cycle to work, although the distance in the high end of this range will become a little more demanding.   If I can do it in 80 minutes or even less each way, that would be ideal. I'm 49 years old, 6 foot tall, and weigh 170lb, and I want to commute by bike because otherwise I'd be getting no exercise in my week at all.  I've never bought or ridden anything but a cheap bike.  What equipment should I buy then, to commute 11 miles each way through a city (Toronto) on  flat, paved roads, shared with cars:   Wheels? Tyres? Clipless pedals and shoes? Gears? Handlebars (e.g. flat or dropped)? Frame (size, shape, material)? Mud-guard/fender? Why do diamonds shine? What is the advantage of accepting the axiom of choice over other axioms (for e.g. axiom of determinacy)?  It seems that there is no clear reason to prefer over other axioms..  Thanks for help. I'd like to update the label of an operator when it's called. But this :  def execute(self,context):     self.bl_label = "good night"   ... apparently doesn't work if the bl_label property has already been defined. Am I wrong ? Nothing shows up in the terminal when I type my password When I'm about to install a program in the terminal it wants the password:  [sudo] password for xxx:   But when I start to type my password nothing happens. What should I do?