If $X=\dbinom m Y \left(\dfrac 1 2\right)^Y \left(\dfrac 1 2\right)^{m-Y}$, and $Y$ has a Binomial distribution with parameters $m$ and $p$, What can we say about the distribution of $X$?
If $X$ follows a Binomial distribution with parameters $n$ and $p$, what can we say on the distribution of $n\choose x$ and $x\choose n$ ? ($x$ is realization of $X$).
Objects, shadows and reflections not rendering in EEVEE Floating sidebar clash with footer in team page Here is a screenshot from  page, with the floating sidebar: (all good so far)        After I scroll down to footer it just combines both:        Can it please be fixed? How to show textures in the 3D View editor? I have added some textures to objects in blender but the textures don't show up until I render the file. it can be difficult to know what the objects look like without rendering it. is there a way to have textures show in the editor even if they are at a lower res?  In editor:    Rendered: solve $T(n)=T(n-1)+T(\frac{n}{2})+n$ Using the recursion tree i tried solving this: $T(n)=T(n-1)+T(\frac{n}{2})+n$; the tree has two parts (branches) one that of $T(n-1)$ and other branch is of $T(\frac{n}{2})$. But as the term T(n-1) reach higher depth, I solved that part of the tree. Finally it resulted in O$(2^n)$ is it correct or is it $O(n^2)$? How can we solve it by substitution method? Mathematica not plotting whole logarithmic function Auto confirm when running bash scripts? From :      ‘We all believe, up at the Lodge, Rachael, that he will be freed from suspicion, sooner or later.’    ‘The better I know it to be so believed there, my dear,’ said Rachael, ‘and the kinder I feel it that you come away from there, purposely to comfort me, and keep me company, and be seen wi’ me when I am not yet free from all suspicion myself, the more grieved I am that I should ever have spoken those mistrusting words to the young lady. And yet I— ’   We usually use should have to say that something was not done, despite being desirable. But in this example, the mistrusting words were indeed spoken (earlier in the book). How would one explain the use of the should have construction then?   Would the following be fully equivalent to the above quote:     ‘We all believe, up at the Lodge, Rachael, that he will be freed from suspicion, sooner or later.’    ‘The better I know it to be so believed there, my dear,’ said Rachael, ‘and the kinder I feel it that you come away from there, purposely to comfort me, and keep me company, and be seen wi’ me when I am not yet free from all suspicion myself, the more grieved I am that I have spoken those mistrusting words to the young lady. And yet I— ’ Cause of black artifacts in render? Do I need a Turkey transit visa for a layover? Some questions might be asked on three different sites, each site needing three different answers. For example, a question about a text editor.  So perhaps there should be a way to link the questions on the three different sites. Adding a link manually in the questions is certainly possible, but perhaps a feature officially supporting this will be beneficial. Studying for CISSP these choices seemed limited:     What is the advantage of digital signatures over message authentication codes?         Digital Signature provides integrity verification while message authentication code does not   Digital Signature provides confidentiality while message authentication code does not   Digital Signatures provide authenticity while message authentication code does not   Digital Signatures works faster than message authentication codes      Using the following chart as a reference, I'm leaning towards #2, but none of the choices really fit properly…  Cryptographic primitive | Hash |    MAC    | Digital Security Goal           |      |           | signature ------------------------+------+-----------+------------- Integrity               |  Yes |    Yes    |   Yes Authentication          |  No  |    Yes    |   Yes Non-repudiation         |  No  |    No     |   Yes ------------------------+------+-----------+------------- Kind of keys            | none | symmetric | asymmetric                         |      |    keys   |    keys   Hopefully an expert here can clarify. Filling triangle doesn't seem to work with PGF/TikZ I was wondering if anyone had any clue on why my triangles are not filled in red within the following piece of code:  \documentclass{article} \usepackage{tikz} \usetikzlibrary{calc,3d}  \usepackage[active, pdftex, tightpage]{preview} \PreviewEnvironment[{[]}]{tikzpicture} \newcounter{number}  \begin{document} \begin{tikzpicture}[x  = {(-0.5cm,-0.5cm)},                     y  = {(0.9659cm,-0.25882cm)},                     z  = {(0cm,1cm)}] \begin{scope}[canvas is yx plane at z=0] \foreach \x / \y / \label in {1 / 0.4 / 1, 0.9 / 0.4 / 0, 1 / 0.5 / 1, 0.7 / 0.3 / 0, 0.8 / 0.4 / 0, 0.9 / 0.5 / 0, 1 / 0.6 / 1, 0.7 / 0.4 / 0, 0.4 / 0.4 / 0, 0.6 / 0.4 / 0, 0.5 / 0.4 / 0, 0.8 / 0.5 / 0, 0.9 / 0.6 / 0, 1 / 0.7 / 1, 0.7 / 0.5 / 0, 0.5 / 0.5 / 0, 0.4 / 0.5 / 0, 0.6 / 0.5 / 0, 0.8 / 0.6 / 0, 0.9 / 0.7 / 0, 1 / 0.8 / 1, 0.7 / 0.6 / 0, 0.6 / 0.6 / 0, 0.5 / 0.6 / 0} { \addtocounter{number}{1} \pgfmathparse{\value{number}}\let\k\pgfmathresult \draw (8 * \x, 8 * \y) node[inner sep=0, outer sep = -0.1pt,color=black,fill,circle] (node-\k) {}; };  \foreach \x / \y / \z / \trash in { 4/5/8/0, 9/11/16/0, 9/16/17/0, 11/10/18/0, 11/18/16/0, 10/8/15/0, 10/15/18/0, 8/5/12/0, 8/12/15/0, 5/2/6/0, 5/6/12/0, 2/1/3/0, 2/3/6/0, 17/16/24/0, 16/18/23/0, 16/23/24/0, 18/15/22/0, 18/22/23/0, 15/12/19/0, 15/19/22/0, 12/6/13/0, 12/13/19/0, 6/3/7/0} { \draw[fill=red]  (node-\x) --  (node-\y) --  (node-\z) -- cycle; } \end{scope} \end{tikzpicture} \end{document}) Why do some font commands behave differently? I've always been confused by the differences in behavior of font commands.  For example, we have \mathrm{} and \texttt{}, which can be used like this: \texttt{hello} world.  And then we have \it and \bf which must be used as {\it hello} world. If one uses \it{hello} world, both words are italicized.  Internally, what is the difference here? And was this difference a conscious design decision? If so, why? Why are the energy levels of a simple harmonic oscillator equally spaced? The energy level of a simple harmonic oscillator is $E_n=(n+\frac{1}{2})\hbar\omega$. Is there any physical explanation why these levels are equally spaced ($= \hbar\omega$)?  Maybe this  can be helpful. Is there any way to increase the number of candy made? In Candy Box 2, the candies are being produced at a rate of 1 candy per second. Is there any way to increase the potential amount of candy made at a faster rate? (notice that I am only up to the part where the person with the cellar in their house gives me a map. Why is Ubuntu 14.04 so slow on my laptop? Suggested reading for renormalization (not only in QFT) What papers/books/reviews can you suggest to learn what  "really" is?  Standard QFT textbooks are usually computation-heavy and provide little physical insight in this regard - after my QFT course, I was left with the impression that Renormalization is just a technical, somewhat arbitrary trick (justified by experience) to get rid of divergences. However, the appearance of Renormalization in other fields of physics  approach in statistical physics etc.), where its necessity and effectiveness have, more or less, clear physical meaning, suggests a general concept beyond the mere "shut up and calculate" ad-hoc gadget it is served as in usual QFT courses.   I'm especially interested in texts providing some unifying insight about renormalization in QFT, statistical physics or pure mathematics. The problem is, it seems when I enable a disabled module, the module's schema version in the system table is automatically updated to the latest version supplied by hook_update_N.  So my current module with the schema version -1 is modified to 7106 when I enable the module, and no hook_update_N functions are run.   So when I come to the next step of running database updates, no hook_update_N functions are run because the module thinks it's already the correct version.  The only method I can see is to manually set the module version in hook_enable, however I'll then need to update the module code to remove the hook_enable to make sure it doesn't run if the module is ever re-enabled in the future, so obviously this approach is a bit dirty.  I'm hoping there's a better solution? Abelian group (Commutative group) Prove that if in a group $(ab)^2= a^2 b^2$ then the group is commutative.  I am having a hard time doing this. Here is what I have so far: Proof:  $a^2 b^2= a^1 a^1 b^1 b^1$ =$aa^{-1}bb$ =ebb Hence,$aa^{-1}=e$  I am stuck, I do not know if this is the right process in proving this Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$? $\{\emptyset\}$ is a set containing the empty set. Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$?  My hypothesis is yes by looking at the form of "the superset $\{\{\emptyset\}\}$" which contains "the subset $\{\emptyset\}$".