Hello how to show the following:  Observations $(X_1,X_2,..,X_n,)$ are i.i.d. over $[0,2\pi)$ with density  $f(x,\theta) = \exp[\theta_1\cos x + \theta_2 \sin x -c(\theta)]$ where  $\theta =(\theta_1,\theta_2) \in R^2$  Find an expression for the Fisher information $I(\theta)$ involving integrals and show that it is positive definite for all $\theta \in R^2$  Thank you!
Let $\theta \in R^{n}$. The Fisher Information Matrix is defined as:  $$I(\theta)_{i,j} = -E\left[\frac{\partial^{2} \log(f(X|\theta))}{\partial \theta_{i} \partial \theta_{j}}\bigg|\theta\right]$$  How can I prove the Fisher Information Matrix is positive semidefinite?
Comparison of backup tools consider positive numbers $a_1,a_2,a_3,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$. does the following in-equality holds and if it does then how to prove it  $\left[(a_1+b_1)(a_2+b_2)\cdots(a_n+b_n)\right]^{1/n}\ge \left(a_1a_2\cdots a_n\right)^{1/n}+\left(b_1b_2\cdots b_n\right)^{1/n}$ If i entered "name" in my text box while it saved it must be saved like "Name"   Capital letters in the first letter of Alphabet  It must not show any error messages it must directly save Can't resolve issue: iptables: No chain/target/match by that name Where does gravity get its energy from? I would like to know where gravity gets its energy to attract physical bodies?  I know that the law of conservation states that total energy of an isolated system cannot change. So gravity has to be getting its energy from somewhere, or else things like hydropower plants wouldn't be able to turn the power of the falling water into a spinning rotor.  Just to be clear, Lets create an example:  Lets say we have two objects with equal mass close to each other. So gravity does its job and it pulls each other closer, this gets turned into kinetic energy. This is where I'm lost. According to the law of conservation energy can't be created or destroyed and the kinetic energy comes from the gravitational pull so where does the gravitational pull gets its energy.  If that energy isn't being recycled from some where else then that means you have just created energy, therefore breaking the law of conservation. Why is /tmp a symlink to /private/tmp? Why is /tmp a symlink to /private/tmp on Mac OS X? In other words, why isn't /tmp just a regular directory, like on Linux or BSD? I understand how it works and I don't mind it, I'm just interested in the (historical?) reasoning behind it. Import users from CSV file Is it possible to import users by using CSV file? or something like a LDAP connection to import all users? Scaling vs Offsetting in Quasi-Poisson GLM I  modeled insurance claim frequency assuming a Quasi-Poisson distribution in R. My frequency dependent variable was calculated, in advance of modeling, as the number of claims per underlying exposure. Someone  that it would be incorrect to do this, and that the correct method was to model the observed claim counts and use the underlying exposure as an offset.   Can someone explain the theoretical and practical differences between scaling and offsetting? I ran the code both ways and noticed that the deviance residuals are significantly smaller when using offsets, but otherwise there were no material differences in variable selection.   P.S. Note the similar but still different question . Why can I not place a pregnant woman to work but others work while pregnant? I am playing Fallout Shelter, have 3 kids in my shelter, all going well, thought I would make one more kid with big strength numbers and decent agility/perception (kids take on SPECIAL stats from their parents). All the women who are pregnant work in my shelter, apart from this one. She can not me put to work, can not be moved from the living shelter, can not be moved to a job via the clipboard short cut.  TL;DR One of the NPC females who is pregnant can not be put to work, can not be moved from the living shelter room, she just walks about as if she is carrying a mammoth inside her. I have NOT moved the shelter, have not let the dad die, have not done anything at all. Hello Physics StackExchange!  I will put the TL;DR in the beginning: I need a self contained, relatively hand-holding sequence of textbooks that covers up from the end of Griffith's Intro to QM to relativistic QM/particle physics to QFT.    This will be my first post. I'm a medical physics grad student who majored in biomedical engineering for my undergrad, with serious regrets about not doing math/physics double major in hindsight. Oh well. I've made it my goal to eventually understand QFT.  My background in relevant subjects is Calc 1 to 3 and Diff EQ, 2 part freshman physics, an E&amp;M course that used Griffiths Electrodynamics, a summer of self studying Griffith's QM (got most of the way through). I will also be officially taking a class in QM using Griffith's textbook this next spring. I am aware of the Lagrangian and Hamiltonian approaches to mechanics as well. In addition I have familiarity with special and basic general relativity using tensors and all this. What would be the best way for me to get from where I am now to a strong understanding of QFT autodidactically? I have no aversion to studying pure maths where I need to.  EDIT: My question is much different than simply asking for a handholding explanation of QFT. Generally these textbooks assume a certain amount of physics/math background, and I was asking for a sequence of textbooks leading up to QFT from the end of a specific textbook, Griffith's QM. Did you even read my question? Did you read the thing you linked me to? No cycle containing edges $e$ and $g$ implies there is a vertex $u$ so that every path sharing one end with $e$ and another with $g$ contains $u$ There is a proof in my textbook for the following claim, which doesn't make a whole lot of sense to me.  My annotations are in bold.  Could someone perhaps elaborate on what's going on?  Claim.  If there does not exist a cycle containing edges $e$ and $g$ then there exists a vertex $u \in V (G)$ such that every path in $G$ sharing one end with e and another with $g$ contains $u$.  Proof: The claim trivially holds if $e$ or $g$ is a loop OK, so we assume that neither is. Let $P$ with vertex set $v_1, v_2, . . . , v_k,$ in order, be a path with $e$ joining $v_1$ to $v_2$ and $g$ joining $v_{k−1}$ and $v_k$. Let $f_i \in E(P_i)$ be the edge with ends $v_i$ and $v_{i+1}$. Let $j$ be chosen minimum so that no cycle in $G$ contains $e$ and $f_j$ We can do this because we know that at least the edge $g$ will not create a cycle by assumption, right?. We will show that $u = v_j$ satisfies the claim.  Suppose not. Let $C$ be a cycle containing $e$ and $f_{j−1}$ What if $f_j = e$?  then what cycle? a single vertex? and let $P′$ be a path from an end of $e$ to an end of $f$ Not sure what the book meant by $f$ here, any guesses? avoiding $u$. Choose a subpath $Q$ of $P′$ with one end in $V (C)$ and another in ${v_{j+1}, v_{j+2}, . . . , v_k}$ as short as possible. Then $C \cup Q \cup P$ contains a cycle containing both $e$ and $f_j$, a contradiction. (The last statement requires some case checking.) This ending seems a bit abrupt and non-obvious to me  Thanks for the help How to boot WUBI installed Ubuntu within Windows 8? After Google Play Service update to version 13 I got an error Lead in ice block This is a strange question I would like to have some explanation accompanied with:  We have an ice block, with in it a lead block. This entire block is put into a glass of water, resulting in a rise of water level until the edge of the glass. The block stays floating on the surface. What will happen with the water level when the ice melts?  The answer is that it will drop, but how is this even true?? My thoughts are: - block floats on water, so rho(block) &lt; rho(water) - Archimedes law, the Fg of block = Fg of water pushed away. - Since a smaller volume of water has equal mass (and so Fg) to block, and ice is less dense than water, when it melts, no mass is lost, so it will take in less space??? Please help me with this point of thinking. How do I configure Apache 2 to run Perl CGI scripts? I would like to configure Apache 2 running on Kubuntu to execute Perl CGI scripts. I've tried some steps that I came across by googling, but nothing seems to work.  What is the right way of achieving this? Solve $f(x)f(2x^2) = f(2x^3+x)$ The full question is to solve $f(x)f(2x^2) = f(2x^3+x)$, prove there are at most one solution per degree of $f(x)$ and then find all the solutions. (Here $f(x)$ is assumed to be polynomial.)  So far I have proven that $(x^2+1)^n$ works for even degree polynomials, and I'm pretty near certain there are no odd solutions.  Is the following a valid argument for there only being at most 1 solution per degree? Let $K$ be the degree of $f(x)$ then $f(x)f(2x^2) = f(2x^3+x)$ produces $3k+1$ linearly independent equations with only $k+1$ independent variables. Therefore for any degree $K$ polynomial there can be at most 1 solution.  Finally, is there any elegant way to prove that no odd solutions exist? Looking for title & author - short story - possibly called "Arms Race" - about rediscovery of manual arithmetic short story - possibly called "Arms Race" - about rediscovery of manual arithmetic.   The hero rediscovers manual calculation - e.g., how to do long division with pencil and paper - in a world where everyone has ben using calculators for aeons. Brought before the president to demonstrate this, the great line in the story is the president, after using his calculator to check the hero's answer, says, "My God -- he's right." A great national project to exploit this advance over the enemy is obviously necessary... I want to draw a table and i used the following code  \begin{table}[htbp]  \centering \caption{Ship simulator required inputs} \begin{tabular}{|c|c|} \hline L_{PP},  L_{WL}, L_{OS}, B, T, T_{a}, T_{f}, C_{B}, D_{P}, `\\` Bulbous bow, Number of bosses, brackets, thrusters. &amp; ship particulars `\\` \hline Required speed or propeller rpm, ship course angle &amp; Mission data \\ \hline BN, weather angle, water temperature &amp; Environment \\  \hline \end{tabular}% \label{inputs}% \end{table}%`   the result of this is as shown in figure    How can i complete the right border of the table ??? I used \\ to break the line of first row Notation: What does the tilde below of the expectation mean? I have a Canon 7D, which has a crop factor of 1.6x.  When using a Canon EF-S lens I get the focal length stated on the lens, when using a Canon EF lens I get the focal length times 1.6 when used on the APS-C body. At least that's what I thought.  I just compared two lenses:   Canon EF-S 15-85mm f3.5-5.6 Canon EF 70-200mm f4   .  The top photo is shot with the 15-85, the bottom photo with the 70-200 - both lenses are set to 70mm. Now why do they look the same?  I thought I would get 112-320mm (70mm*1.6 crop factor). What did I misunderstand about the crop factor calculation? Lightroom also displays the photos as having "70mm" and not 112mm.