We found normal distribution is so common in nature, such as many measurement of species (weight, height or size). From the point of central Limit Theorem, Can I intuitively understand this as the additive result of multiple factors (genes affecting height) that affect corresponding phenotype (height)?
In several different contexts we invoke the  to justify whatever statistical method we want to adopt (e.g., approximate the binomial distribution by a normal distribution). I understand the technical details as to why the theorem is true but it just now occurred to me that I do not really understand the intuition behind the central limit theorem.  So, what is the intuition behind the central limit theorem?   Layman explanations would be ideal. If some technical detail is needed please assume that I understand the concepts of a pdf, cdf, random variable etc but have no knowledge of convergence concepts, characteristic functions or anything to do with measure theory.
Is ¬¬(¬¬P → P) provable in intuitionistic logic? How do I give myself an item that can only be placed on a block with a certain damage value? One thing I've noticed about the relatively new sitewide layout is that I can't find a quick way to get back to www.stackexchange.com without finding StackExchange in the sites list. As a result, I am typing stackexchange in my toolbar a whole lot more often now. In particular, I tend to enjoy browsing the hot questions list and my own personal feed quite often so visit these pages on a daily basis. Is there an easy link to them that I'm missing?  (On the android app, there is a feed in the menu that I really appreciate, something like that on the desktop would be ideal!) Need to draw a flowchart as in the image below with some arrows I have problem with the feed back from bottom block to on top of top block. Can anyone help? $$\int x^3\sqrt{4-x^2}dx$$ I tried changing the expression like this: $$\int x^3\sqrt{2^2-x^2}dx=\int x^3\sqrt{(2-x)(2+x)}dx$$ And that's where I got stuck. I tried integration by parts but it doesn't simplify the expression. Becoming Thieves Guild Guildmaster, am I bugged? I've completed the Thieves Guild main questline () and chosen my special power (Shadowcloak of Nocturnal). It says completed in my quest log.  I also completed the  in each city you can give Thieves guild influence to (Markarth, Whiterun, Windhelm, and Solitude) and done their special jobs from Delvin. They show completed in my quest log. I've even done at least one of each kind of job from Vex and Delvin (but not in each city).  I got  in the Ragged Flagon and have done the quest for Tonilia (delivering the Moon Sugar to a caravan) and Vekel (getting the 4 books). I've also upgrade one of my pieces of Thief Armor.  I talk to  to become guild master and I get no dialogue options but he just says he's busy and we'll talk another time. There is a guild chest in front of his desk but it's locked and I don't have the key.  What am I missing? Is there some bug I've encountered? Accurately simulating the lots of dice rolls without loops? OK so if your game rolls lots of dice you can just call a random number generator in a loop. But for any set of dice being rolled often enough you will get a distribution curve/histogram. So my question is there a nice simple calculation I can run that will give me a number that fits that distribution?  E.g. 2D6 - Score - % Probability  2 - 2.77%  3 - 5.55%  4 - 8.33%  5 - 11.11%  6 - 13.88%  7 - 16.66%  8 - 13.88%  9 - 11.11%  10 - 8.33%  11 - 5.55%  12 - 2.77%  So knowing the above you could roll a single d100 and work out an accurate 2D6 value. But once we start with 10D6, 50D6, 100D6, 1000D6 this could save a lot of processing time. So there must be a tutorial / method / algorithm that can do this fast? It is probably handy for stock markets, casinos, strategy games, dwarf fortress etc. What if you could simulate the outcomes of complete strategic battle that would take hours to play with a few calls to this function and some basic maths? Motivation behind standard deviation? Enumeration within self defined environment My self defined environment having a symbol at the beginning and at the end works fine for text without enumeration. When including enumerate the symbols are not there where desired. Is it possible to have the first symbol directly in front of &quot;1.&quot; i. e. in the same line and the ending symbol in the same line as the last line of the enumeration? With \ignorespacesafterend and \unskip I did not succeed. \documentclass{article} \usepackage{amssymb} \newenvironment{myenv}{$\blacktriangleright$}{\hfill$\blacktriangleleft$}  \begin{document} There is a text.  \begin{myenv} And an example \end{myenv}  Here is another text.  \begin{myenv} \begin{enumerate} \item A \item B \end{enumerate} \end{myenv}  \end{document} Get the exact time for a remote server /usr/bin/sudo must be owned by uid 0 and have the setuid bit set I have run following command accidentally   sudo chown [username] -hR /   Now sudo su getting error:  sudo: /usr/bin/sudo must be owned by uid 0 and have the setuid bit set   How to Solve This? Where does the curve $x^y = y^x$ intersect itself? This problem is quite easily solved by using logarithms and derivative and forming the function $f(x,y) = x^y - y^x$, however there are assertions that this problem can be solved without using either of the two. I can not see how one would proceed to solve this problem in the absence of logarithmic manipulation and differentiation. Can anyone shed some light on this? File list command line (hidden and subfolders) I need to get a text list of all the files (hidden and visible) off a hard drive, subfolders included. Ideally the list would have the filename, path, size, and creation (or last modified) date if possible. Can someone please tell me what command I need? Also, is it possible to have this created as a .csv file or something similar for use in Excel?  I am not super proficient with Ubuntu so an explanation of each command item would also be greatly appreciated. Convergence of sequence given by $x_1=1$ and $x_{n+1}=x_n+\sqrt{x_n^2+1}$ Be $(x_n)_n\ge 1 $  such that $x_1=1$ and $x_{n+1}=x_n+\sqrt{x_n^2+1}$ for every $n\ge 1$. Prove that the sequence $y_n=(2^n/x_n)_n \ge 1$ is convergent and find it s limit.  Being positive and decreasing, $y_n$ is clearly convergent. But finding it's limit really put me intro trouble. Any help?  Thank you! What is Peter Quill's status after Guardians of the Galaxy Vol. 2? Prevent Serialization of Base Class Let $m,n$ be positive integers bigger than $1$. Show that $\mathbb{Z}_m \times \mathbb Z_n$ is cyclic if and only if $\gcd(m,n)=1$.    I have no idea on how to start. Anyone hints are much helpful. Can't replace block with a Command Block I am making a Sell Shop, but I need to replace some things using /setblock.  For one of the replacements, I am using the command:  /setblock 579 62 1053 minecraft:command_block 0 replace {Command:"testfor @p {Inventory:[{id:"minecraft:log",Count:32b}]}",TrackOutput:0}  But it won't work! When I try it using a Command Block, I get this message:     [19:14:38]   Data tag parsing failed: Unexpected token 'm' at: minecraft:log",Count:32b}]}",TrackOutput:0   Is there any way I can fix this? Alternate moves, first one to roll a 6 wins: what's the probability of winning? Question: Players A and B play a sequence of independent games. Player A starts and throws a fair die and wins on a "six". If she fails then player B throws the same die and wins on a "six" or fails otherwise. If she fails then player A gets the die and throws again, and so on. The first player to throw a "six" wins. Find the probability that player A wins.  Here is what I have done:  A wins right away: $\dfrac{1}{6}$  A fails B wins: $\dfrac{5}{6}\times \dfrac{1}{6}$  A fails B fails A wins: $\dfrac{5}{6}\times \dfrac{5}{6}\times \dfrac{1}{6}$  Unlike another similar question which the player B can win on "five" and "six" after player A fails on the first throw, in this case, the games do not seem to have an endpoint. How should I calculate the probability that player A wins then? Why does Python pep-8 strongly recommend spaces over tabs for indentation?