Consider a non-symmetric coin experiment where $p$ is the probability to get heads. I flip the coin $n$ times and try to estimate the value of $p$. The amount of heads has the binomial distribution so the Fisher information for the distribution parameter $p$ is $$I_p = \frac{n}{p(1-p)}$$ and the variance for an effective estimator is $$D_p = \frac{p(1-p)}{n}.$$ When $p$ is very small one got a very high information and low variance so I need much smaller sample size to get the same accuracy as for the case $p=1/2$.  What is the intuition behind that? Why am I almost certain about the value of $p$ after few throws if it is small?
It confuses/blows my mind that the Binomial has variance proportional to $p(1-p)$. Equivalently, the Fisher information is proportional to $\frac{1}{p(1-p)}$. What is the reason for this? Why is the Fisher Information minimized at $p=0.5$? That is, why is inference most difficult at $p=0.5$?  Context:  I'm working on a sample size calculator, and the formula for $N$, the sample size needed, is an increasing factor of $p(1-p)$, the result of a variance estimation in the derivation.
Why are map tiles being repeated? Using MapServer and Openlayers I accidentally killed the orphanage lady before I started the quest. She just pissed me off, sadly it will not acknowledge that I have finished the quest and I haven't been contacted. Is there another way to join the Dark Brotherhood? Will getting a Software Testing Certification benefit me in my career? Why does the natural ring homomorphism induce a surjective group homomorphism of units? I'm trying an old problem here:   Suppose $n\mid m$, and I have a natural ring homomorphism $\varphi\colon \mathbb{Z}/m\mathbb{Z}\to\mathbb{Z}/n\mathbb{Z}$ defined by $\varphi(j)=j\mod{n}$. I can verify that this is a ring homomorphism, but why does it induce a surjective group homomorphism on the unit groups $(\mathbb{Z}/m\mathbb{Z})^\times\to(\mathbb{Z}/n\mathbb{Z})^\times$?  My question in particular is why is it surjective? I take $k\in(\mathbb{Z}/n\mathbb{Z})^\times$. Then there exist integers $s,t$ such that $sn+tk=1$. Since $n\mid m$, I can also write $m=na$. So multiplying through I get $sm+tka=a$. I'm lost here. How can I find a unit in $(\mathbb{Z}/m\mathbb{Z})^\times$ which maps to $k$ to show surjectivity? How to chat to all I very rarely see chats too all and don't know how to do it myself. Normally it's SHIFT + ENTER The following hotkeys have been tried: - SHIFT + ENTER - CTRL + ENTER - ALT + ENTER What point (pt) font size are \Large etc.? In the preamble I have:  \documentclass[a4paper,11pt]{article} \usepackage{fontspec} \setmainfont{Arial}   What can be inferred about the real font pt size for the following?  \tiny, \scriptsize, \footnotesize, \small, \normalsize, \large, \Large, \LARGE, \huge, \Huge Update 02/02/2019 I appreciate all the responses and help but unfortunately i must have not seen the emails stating there were responses! I am now looking into a NEST thermostat and was curious if they too, use the same connections. Lastly, looking for a quick recommendation on which WiFi Compatible therms are preferred by the memebers that respond! Thanks again your knowledge is highly appreciated! Bob  I Purchased the Honeywell WiFi Prog Thermostat to replace my Round Digital Honeywell Thermostat. I have separate Heating and Cooling (Heating is a Weil McLain Boiler for Hot water baseboard and Hot water(no tank) and Cooling is a separate unit that was added later.)  Wires:  RC Cool R Heat W Heat Y Cool G Cool   3 wires from AC unit and 2 from Heater.  If I were to go the separate transformer route to power the thermostat, where would I land the 2 wires?  C and ??? String Comparison in Java Calculate the area on a sphere of the intersection of two spherical caps Where's the error in this $2=1$ fake proof? I'm reading Spivak's Calculus:  2 What's wrong with the following &quot;proof&quot;? Let $x=y$. Then $$x^2=xy\tag{1}$$ $$x^2-y^2=xy-y^2\tag{2}$$ $$(x+y)(x-y)=y(x-y)\tag{3}$$ $$x+y=y\tag{4}$$ $$2y=y\tag{5}$$ $$2=1\tag{6}$$  I guess the problem  is in $(3)$, it seems he tried to divide both sides by $(x-y)$. The operation would be acceptable in an example such as:  $$12x=12\tag{1}$$ $$\frac{12x}{12}=\frac{12}{12}\tag{2}$$ $$x=1\tag{3}$$  I'm lost at what should be causing this, my naive exploration in the nature of both examples came to the following: In the case of $12x=12$, we have an imbalance: We have $x$ in only one side then operations and dividing both sides by $12$ make sense. Also, In  $\color{red}{12}\color{green}{x}=12$ we have a $\color{red}{coefficient}$ and a $\color{green}{variable}$, the nature of those seems to differ from the nature of $$\color{green}{(x+y)}\color{red}{(x-y)}=y(x-y)$$ It's like: It's okay to do the thing in $12x=12$, but for doing it on $(x+y)(x-y)=y(x-y)$ we need first to simplify $(x+y)(x-y)$ to $x^2-y^2$. Possible Duplicate:          Here is my sync script:  ssh user@remote.host /etc/init.d/some_service stop scp user@remote.host:/var/some_service/events ./events scp ./new_data user@remote.host:/var/some_service/new_data ssh user@remote.host /etc/init.d/some_service start   As You can see, it stops service, perform some copies with scp and starts service back.  The problem: it "costs" 4 "similar" ssh connections in a row.  Is there any way to make all the work, using one physical ssh connection? Does assigning objects to null in Java impact garbage collection? Does MariaDB or MySQL implement the VALUES (expression) table value constructor? I am just wondering if MariaDB or MySQL implements the &lt;table value constructor&gt; in the SQL Spec. In  and ?  SELECT * FROM ( VALUES (1) ) AS t(x);  x  ---  1 (1 row)   (syntax from ). Java; String replace (using regular expressions)? As part of a project for school, I need to replace a string from the form:  5 * x^3 - 6 * x^1 + 1   to something like:  5x&lt;sup&gt;3&lt;/sup&gt; - 6x&lt;sup&gt;1&lt;/sup&gt; + 1   I believe this can be done with regular expressions, but I don't know how to do it yet.  Can you lend me a hand?   P.S. The actual assignment is to implement a Polynomial Processing Java application, and I'm using this to pass polynomial.toString() from the model to the view, and I want do display it using html tags in a pretty way. Division rules for other number systems? How could we make the same division rules for other number systems, like in our decimal system: a number is divisible with 2 if it's last digit is 0,2,4,6,8, by 3 if the sum of digits is divisible with 3, and so on.  Can we do the same rules for g-based number systems too, where g is not 10, and greater than 1? Which is better: side quests after killing the warrior or Playthrough 2? My question is, once I beat the warrior should I do all the sidequests in Normal before starting TVHM or should I go right to TVHM once it becomes available?    I just bought the game and have been playing with some friends who are higher level than me so that we could enjoy the Tiny Tina DLC together.  As a result, I leveled up pretty fast without doing much of the main story quest or side quests.  Now I'm a level 32 commando and haven't finished the main storyline in normal (Just finished "Where angels fear to treat Pt 2").  I get almost no XP from kills on missions.  I still have tons of sidequests.  I imagine some of them might give good loot, or be fun to play.  It it better to go to Playthrough 2 or should i do the sidequests in Normal first?  EDIT: changed question title and some wording for clarity How do I show that there exists variational/action principle for a given classical system? We see variational principles coming into play in different places such as Classical Mechanics (Hamilton's principle which gives rise to the Euler-Lagrange equations), Optics (in the form of Fermat's principle) and even General Relativity (we get Einstein's equation from the Einstein-Hilbert action). However, how do we explain this very principle, i.e., more mathematically, I want to ask the following:      If I am given a set of generalized positions and velocities, say, $\{q_{i}, \dot{q}_{i}\}$, which describes a classical system with known dynamics (equations of motion), then, how do I rigorously show that there always exists an action functional $A$, where    $$A ~=~ \int L(q_{i}, \dot{q}_{i})dt,$$    such that $\delta A = 0$ gives the correct equations of motion and trajectory of the system?    I presume historically, the motivation came from Optics: i.e., light rays travel along a path where $S = \int_{A}^{B} n ds$ is minimized (or at least stationary). (Here, $ds$ is the differential element along the path). I don't mind some symplectic geometry talk if that is needed at all. Prove that the product of a non-zero rational and irrational number is irrational. Could you please confirm if this proof is correct?  Theorem: If $q \neq 0$ is rational and $y$ is irrational, then $qy$ is irrational.  Proof: Proof by contradiction, we assume that $qy$ is rational. Therefore $qy=\frac{a}{b}$ for integers $a$, $b \neq 0$. Since $q$ is rational, we have $\frac{x}{z}y=\frac{a}{b}$ for integers $x \neq 0$, $z \neq 0$. Therefore, $xy = a$, and $y=\frac{a}{x}$. Since both $a$ and $x$ are integers, $y$ is rational, leading to a contradiction. The only thing I need is an exact copy of an existing shapefile in the same directory with a different name.  I have tried:   arcpy.CopyFeatures_management (ORLinks, MyLinks) #ORLinks: old links, MyLinks: new links   but for some reason, I cant make it a feature layer using:  arcpy.MakeFeatureLayer_management(MyLinks, 'MyLinks')   Error Encountered:  Error: Failed to execute. Parameters are not valid. ERROR 000732: Input Features: Dataset  M:\RAIL\Rail2.0.shp does not exist or is not supported Failed to execute (MakeFeatureLayer). SharePoint People.aspx Page is Visible for All Users The SharePoint People.aspx page is visible for all users by default. Is there any way to restrict this page from certain readers?