I'm trying to figure out the difference between an A/B split test and an ANOVA test and I came across this . It suggests that the mean of $n$ independent Bernoulli random variables is normally distributed. Is this true?  If I have a random variable $y$ that follows a Bernoulli distribution:  $F(\theta) = \theta^{y}(1-\theta)^{1-y}$  And it takes values of 1 or 0, will the mean of $n$ of these variables be normally distributed?  $\frac{1}{n}\sum y_{i}$
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.
To prove Cayley-Hamilton theorem, why can't we substitute $A$ for $\lambda$ in $p(\lambda) = \det(\lambda I - A)$? If HTML, CSS, and Javascript are client-side, why are they components of a PHP file? If $\lambda_{max} = n$, then $n\times n$ positive, reciprocal matrix A is consistent At the end of . the author states that  Suppose we have $n \times n$ matrix $A$ having only positive elements and satisfying the property $a_{ij}=1/a_{ji}$ (a matrix satisfying this property is called a reciprocal matrix).  If its largest eigenvalue $\lambda_{max}$ is equal to $n$, then the matrix $A$ satisfies the property (consistency property) $a_{ij} a_{jk} = a_{ik}$ where $i,j,k=1,2,...,n$.  I'm not convinced this theorem is true. Could anyone help? How to find the center of an ellipse? What is the difference between $Q=\Delta U+W$ and $\Delta U=Q+W$? Does anyone know where I can find cheat codes for Goldeneye 007 Wii? I know that the game hasn't even been out for a week, but does anyone know if any cheat codes have been released by Activision yet?  I know that they must exist since there is a cheat codes menu within the game. Opening a file with std::string Use LaTeX to simulate old typewriter written texts This question led to a new package:      I'm giving my students an old exam and barely remembered to change the date at the top. A colleague and I joked about how long we could reuse the same exam, and it made me wonder if I could use LaTeX to create an exam that looked like it was written in 1963 on a typewriter.  Here's what I'd like to see:   Typewriter font even in math mode. (Several ways to accomplish this; is there a "best" way?) Slight inconsistencies in the printing of each letter. I attempt this below adapting code taken from . Each word has a random vertical shift, though I'd like to see it reduced to each letter having its own vertical shift. Other typewriter-like irregularities would be great. It would be great if math-mode delimeters \left(, \right}, etc., can be created in a piece-meal fashion as they did in days of old. Fraction bars, tops of square-root symbols -- any horizontal line, really -- is made up of a series of en-dashes that don't quite match up.   Here is what I've come up with so far:  \documentclass[10pt]{article}% This is a document class providing more font size options \usepackage{graphicx} \usepackage{ifthen} \usepackage[textwidth=7.5in,textheight=9.5in]{geometry} \pagestyle{empty}  % thanks to Bruno Le Floch: https://tex.stackexchange.com/q/9331/4012 % and in his comments to https://tex.stackexchange.com/a/29458/4012 \usepackage[first=-1,last=1]{lcg}% you can play around with these values \makeatletter \newcommand{\globalrand}{\rand\global\cr@nd\cr@nd} \makeatother  \newcommand{\randomvshift}[1]{\globalrand\raisebox{\value{rand}pt}{#1}}  %%% thanks to Martin Scharrer: https://tex.stackexchange.com/q/11598/4012 \makeatletter \def\typewriter#1{%     \@typewriter#1 \@empty } \def\@typewriter#1 #2{%    \randomvshift{#1}\space    \ifx #2\@empty\else     \expandafter\@typewriter    \fi    #2% } \makeatother \begin{document}\tt  \typewriter{Math 101 Fall 1963}  \begin{enumerate} \item   \typewriter{$\frac{\texttt{d}}{\texttt{dx}}\texttt{(x)}^2$}  \item \typewriter{State the Fundamental Theorem of Calculus.}  \end{enumerate}  \end{document} We know that Accio spell (aka ) has some limits:   It can be blocked (e.g. Harry couldn't Accio the Diadem in RoR in Deathly Hallows). Many more failures are listed on Wikia.   But what are the exact full rules and limitations governing the spell in Harry Potter world?   Distance?  Line-of-sight?  Knowing exact or approximate location?  Specificity of item (e.g. "Accio Broom" seemed to work for Weasleys in OotP to bring their locked-up brooms - why not just a random broom?) Types of Accio-able stuff? Can you Accio water? Lump of earth? Gases/liquids/etc...? Can you Accio based on jQuery selector (or an SQL where clause), for lack of a better description? E.g. "Accio the book that has the word 'Flammel'"? "Accio biggest green ball from this room"? Can you Accio ALL items of a kind? (the reason I thought of this whole question was seeing "Accio all guns" comment in ). Can you Accio vague stuff? ("Accio meal"). Other rules that can be inferred.   NOTE: While Wikia article has some of the info, it is clearly very lacking in details on specific rules based on specific instances in canon. I'm looking for a fully comprehensive, organized ruleset for the charm, not merely 1-2 tidbits in Wikia article. Mutable or immutable class? What is the purpose of the word 'self'? Currently, when someone suggests an edit, they are unable to keep track of it without watching reputation levels or constantly refreshing the page. What I would like is a notification, that shows up like the notification where someone has edited your question, that shows whether a suggested edit has been approved or rejected after the last vote is cast. This way, the community can be notified when their edit has been rejected/approved, while focusing their efforts on another question/answer. Can I attack a specific player? I was in a clan and someone kicked me from it.   I think I'm stronger than him. Is there any way I can attack him? The only method I see to attack is random players. Can I attack a specific player? round in PHP shows scientific notation instead of full number I'm trying to do an echo of a variable containing 1400000.  so there is written: echo round(1400000); this gives 1,4E+6 instead of the full number. Anybody an idea on how to display it fully? How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$? Nesting HTML- anchor tags For , the answer given on 7 Feb about how to edit the subject line in the new format of Gmail does not work for me.   There is no box on the left with a down arrow to options to reply, forward or edit subject in my version. There is a book on the right offering options to    Reply Reply to all Forward Filter message Print Delete Report spam Report phising Show original Message text garbled Translate message Mark answered    But, no option to edit subject How to Fix Broken Packages? How do I fix broken packages when it is not working? When I try to fix broken packages, it tells me to "fix broken packages." Unexpected gaps in IDENTITY column I'm trying to generate unique purchase order numbers that start at 1 and increment by 1. I have a PONumber table created using this script:  CREATE TABLE [dbo].[PONumbers] (   [PONumberPK] [int] IDENTITY(1,1) NOT NULL,   [NewPONo] [bit] NOT NULL,   [DateInserted] [datetime] NOT NULL DEFAULT GETDATE(),   CONSTRAINT [PONumbersPK] PRIMARY KEY CLUSTERED ([PONumberPK] ASC)     );   And a stored procedure created using this script:  CREATE PROCEDURE [dbo].[GetPONumber]  AS BEGIN     SET NOCOUNT ON;      INSERT INTO [dbo].[PONumbers]([NewPONo]) VALUES(1);     SELECT SCOPE_IDENTITY() AS PONumber; END   At the time of creation, this works fine. When the stored procedure runs, it starts at the desired number and increments by 1.  The strange thing is that, if I shut down or hibernate my computer, then the next time the procedure runs, the sequence has advanced by almost 1000.   See results below:    You can see that the number jumped from 8 to 1002!   Why is this happening?  How do I ensure that numbers aren't skipped like that?  All I need is for SQL to generate numbers that are:   a) Guaranteed unique.  b) increment by the desired amount.    I admit I'm not a SQL expert. Do I misunderstand what SCOPE_IDENTITY() does?  Should I be using a different approach? I looked into sequences in SQL 2012+, but Microsoft says that they are not guaranteed to be unique by default. How can I make my microSD card writable?