Given a sample $ X_1,X_2 \dots X_{100}$ and the density function   $ f(x;\theta) = \frac{1}{\pi \cdot \left(1+\left(x-\theta \right)^2\right)}$ , find Fisher information for  $\hat{\theta}_{MLE.}$ My attempt: $L(\theta;x_1,x_2\dots x_{100}) = \prod _{i=1}^{100}\left(\frac{1}{\pi \cdot \left(1+\left(x_i-\theta \right)^2\right)}\right)\:$ $l$ = $\log(L) = -100*\ln(\pi)-\sum^{100}_{i=1}(\ln(1+(x-\theta)^2)$. $\frac{\partial }{\partial \theta}\left(\log(L)\right) = \sum_{i=1}^{100}(\frac{2(x_i-\theta)}{1+(x_i-\theta)^2}$ $\frac{\partial^2 }{\partial^2 \theta}\left(\log(L)\right) = \sum_{i=1}^{100}-\frac{2\left(-θ+x_i\right)}{1+\left(x_i-θ\right)^2}$ fisher's information : $I(\theta) = -E[\frac{\partial^2 }{\partial^2 \theta}\left(\log(L)\right)$ but I don't how to find the expected value, I tried to simplify it but that didn't help.
I need to calculate the Cramer Rao lower bound of variance for the parameter $\theta$ of the distribution $$f(x)=\frac{1}{\pi(1+(x-\theta)^2)}$$   How do I proceed I have calculated $$4 E\frac{(X-\theta)^2}{1+X^2+\theta^2-2X\theta}$$  Can somebody help
$\operatorname{Ext}$ and injectives, respectively projectives Why should we typedef a struct so often in C? What does "rigorous proof" mean? I have heard several times that some mathematician has given another and more rigorous for an established theorem, but I don't know what does it really mean and what differences makes it to be more 'rigorous'. My understanding of a proof is that a proof is some explanation to convincing others that a statement is true. If some proof is not rigorous, does it mean that while the proof is acceptable for many but it is not for everybody? If so, the proof is not correct since it going to be rejected by some people and it is not a proof at all. Or, does rigorous mean it uses more advanced tools? If so, why not any alternative (but simpler) proof is also acceptable as rigorous (which is more 'valuable' as well since it's simpler!)?   Saying those limited insight of mine, my questions are:   $1-$ What does rigorous proof mean, in general definition?  $2-$ In order to understand your answers, as an example I am re-writing three of the proofs for the compactness of the torus. Which of the following proofs are most rigorous, 'normal' and least rigorous; and why so?    Theorem: The torus $T^2$ is compact.  First proof: The torus is homeomorphic to the product space $S^1 \times S^1$. The circle $S^1$ is compact. Therefore the torus is the product of two compact spaces, and thus it is compact.   Second proof: The torus $T$ is the subspace of $\mathbb{R}^3$ obtained by rotating a circle about the $z$-axis. The torus is closed since given any point in its complement, there is an open ball of sufficiently small radius centered at that point and disjoint from the torus. Also, the torus is bounded since it is contained in the ball of radius $4$ centered at the origin. Since the torus is closed and bounded in $\mathbb{R}^3$, it is compact.   Third proof: The torus is homeomorphic to a space obtained as a quotient  space of the square by identifying opposite edges. The square is a closed and  bounded subset of $\mathbb{R}^3$ and therefore is compact. Furthermore, a quotient map on the square is a continuous function, and the image of a compact space under a continuous function is compact. Therefore the torus is compact.   Thanks a lot for your guidance. Word to describe someone who goes to all the events in town! Is Angular Momentum truly fundamental? This may seem like a slightly trite question, but it is one that has long intrigued me.  Since I formally learned classical (Newtonian) mechanics, it has often struck me that angular momentum (and generally rotational dynamics) can be fully derived from normal (linear) momentum and dynamics. Simply by considering circular motion of a point mass and introducing new quantities, it seems one can describe and explain angular momentum fully without any new postulates. In this sense, I am lead to believe only ordinary momentum and dynamics are fundamental to mechanics, with rotational stuff effectively being a corollary.  Then at a later point I learned quantum mechanics. Alright, so orbital angular momentum does not really disturb my picture of the origin/fundamentality, but when we consider the concept of spin, this introduces a problem in this proposed (philosophical) understanding. Spin is apparently intrinsic angular momentum; that is, it applies to a point particle. Something can possess angular momentum that is not actually moving/rotating - a concept that does not exist in classical mechanics! Does this imply that angular momentum is in fact a fundamental quantity, intrinsic to the universe in some sense?  It somewhat bothers me that that fundamental particles such as electrons and quarks can possess their own angular momentum (spin), when otherwise angular momentum/rotational dynamics would fall out quite naturally from normal (linear) mechanics. There are of course some fringe theories that propose that even these so-called fundamental particles are composite, but at the moment physicists widely accept the concept of intrinsic angular momentum. In any case, can this dilemma be resolved, or do we simply have to extend our framework of fundamental quantities? Probability that the bag contains all balls white given that two balls are white A bag contains $4$ balls .Two balls are drawn at random and are found to be white.  What is the probability that all the four balls in the bag are white?.  The options are $\large\frac{1}{5}\space ;\frac{2}{5} ;\frac{3}{5} ;\frac{4}{5}$  My work: It's given that two balls are white.So the rest of the balls could be    all white One of them white none of them white   So among the three above possibilities only one satisfies our conditions so the probability is $\frac{1}{3}$.But it not even the options given. What's wrong with my work? Let $f:G\to K$ be a surjective group homomorphism and $H$ be a subgroup of $G$ satisfying $N_G(H)=H$. Then $N_K(f(H))=f(H)$. I'm trying to prove or disprove it. perhaps, this proposition is true. But, how can we prove it? Edit: I thought it is true because WIKI said it is true without proof Let $f\colon(0,\infty)\to\mathbb R$ be differentiable on $(0,\infty)$ and suppose that $\lim\limits_{x\to\infty} f'(x)=L$.         a. Show that for any $h&gt;0$, $\lim\limits_{x\to\infty} \frac{f(x+h)-f(x)}h=L$.      b. Show that $\lim\limits_{x\to\infty} \frac{f(x)} {x} =L$.   this is a differentiation question.  i don't know how to solve this question  Because it is Korean, it is difficult to deal with this cite. So for me, it is not easy to write.  thank you I'm looking for one word which describes what a student who missed classes needs to do Combine rows / concatenate rows Complicated FOR loop nesting. Having trouble cleaning up legacy code RC4 , Is it possible to find the key if we know the plaintext and ciphertext? Is it possible to find the key if we know the plaintext and ciphertext with RC4. How should I write the algorithm? Irreducibility check for polynomials not satisfying Eisenstein Criterion. My Question is to check Irreducibility for polynomials not satisfying Eisenstein Criterion.  As an Illustration, to check whether $x^{p-1}+.....+x+1$ for p a prime is irreducible or not, we replaced $x$ by $x+1$ and by using Eisenstein's Criterion for the resulting polynomial we conclude that resulting polynomial is irreducible and so is the original polynomial.  In general, for a given irreducible polynomial $f(x)$ with coefficients in a known U.F.D, is there some element $a$ such that we can apply Eisenstein's criterion to $f(x+a)$?  I am sure there would be no general structure for this but I expect there to be at least some special cases.   Any Reference/suggestion would be appreciated.  Thank You. Solution to German Tank Problem Is there a formal mathematical proof that the solution to the  is a function of only the parameters k (number of observed samples) and m (maximum value among observed samples)? In other words, can one prove that the solution is independent of the other sample values besides the maximum value? I would like to take a picture of a street during the day to depict how much traffic and cars there are on the street.  Is there anyway to do this with a long exposure photograph?  I have a Nikon D90 and I set the f stop down to 1/16 (the lowest of my lens) and set the time to 2 seconds. Doing this, the exposure is almost saturated when taken around 5pm with the sun setting behind a tree. Doing this, the cars passing by were very hard to make out against the stationary background, being visible as a blur in the color of the car.  Perhaps this is not possible and I should just do this at night where the cars lights are much brighter than the background.  Please suggest how else I might depict the traffic in a photograph?  Are daytime long exposure photographs even possible? There's a problem statement:     For each $m \in \mathbb{N}$, we construct a sequence $m_0$, $m_1$, $m_2,\dots$ denoted $S_m$, recursively via $m_0=m$ and      $$m_{i+1} = m_i + \left\lfloor \sqrt{m_i} \right\rfloor$$      for all $i \ge 0$.   Here, $\lfloor x \rfloor$ is the floor of $x$, the greatest integer less than or equal to $x$.   Hence, we have $\left\lfloor \sqrt{10} \right\rfloor=3$ and $\left\lfloor \sqrt{29} \right\rfloor=5$.      Show that for each positive integer $m$, the sequence $S_m$ contains the square of some integer.   I'm pretty certain that this can be proved with induction. I am just not quite sure what to induct on. Examining examples shows that $S_m$ always results in a perfect square eventually, though I'm not sure how to prove it. Photoshop CC 2019: Pixelated Brush Stroke For some reasons my brush stroke suddenly looks like this:     As the picture showed, the edges are jagged and pixelated. It doesn't happen before and it's just one of the built-in inking brush.  I know there are a lot of similar questions here but please don't mark this as duplicate. I double checked all the thing the other answers suggested: It's brush tool, not pencil, and my mode of brush and layer is Normal, not Dissolve. It's pixelated even if I draw with mouse so it's not tablet driver issue. I disabled GPU render but nothing changed. How do I fix this? I would like to know how to unload an assembly that is loaded into the main AppDomain.  I have the following code:  var assembly = Assembly.LoadFrom( FilePathHere );   I need/want to be able to unload this assembly when I am done.  Thanks for your help. Pre-image of a measurable set A is always measurable? Let $f$ be a continuous function on a set $E$.  Is it always true that $f^{-1}(A)$ is always measurable if $A$ is measurable?  I say no.  We know that $\bar{\psi(x)}:=\frac{\phi(x)+x}{2}$ where $\phi$ is the Cantor (or Cantor-Lebesgue) function and $\bar{\psi}:[0,1] \rightarrow [0,1]$.  We know this function maps a measurable subset of $C$ (the Cantor set) into a non-measurable set $W$.  A few observations about $\bar{\psi(x)}$.  It is strictly increasing and continuous, so it has a continuous inverse $\bar{\psi(x)}^{-1}$.  Thus $\bar{\psi(x)}^{-1}(W)=c \subset C$; that is, $\bar{\psi(x)}^{-1}(W)$ is non-measurable, but c is.  Is this correct? What is the difference between correlation, causation and prediction? Suppose we have a set of events $\Omega$, containing events $A$ and $B$. My econometrics professor tried to distinguish the following three terms today.    Causation --- $A$ causes $B$ if the occurrence of $A$ always leads to another specific outcome $B$. For example, clapping my hands causes a sound to be emitted.  Prediction --- $A$ predicts $B$ if on average, $B$ is the expected outcome from $A$ occurring. In other words, whereas causality is deterministic, prediction is probabilistic. For example, studying for a test would predict doing well on the exam. But there's no guarantee.   Correlation --- $A$ and $B$ are correlated if when one occurs the other does too. But there may be some hidden variable causing both.    Are these distinctions meaningful? Or did he make these terms up?   My Questions   What is the difference between causation and correlation?  Are "causes" always deterministic?