Suppose we have $T_i,i=1..n$ i.i.d. with unknown distribution and we want to estimate $E[T]$. Note that in this setting we are not estimating E[T] as a parameter of a parameter-dependent family of distributions, therefore it is difficult to attach a meaning to a likelyhood function:  $L[E[T]]=P(T_i|E[T])$  as it would be done when for example estimating $\mu$ from maximum likelyhood, knowing that the underlying distribution is Gaussian.  Again what we would probably do is computing $\overline{T}=\frac{1}{n}\sum_i T_i$. And we would be sure that the estimator would be consistent and its variance would go to zero. For example we have trivially:  $E[\overline{T}]=E[T_i]$  and:  $\sigma^2[\overline{T}]=O(1/n)$  supposing that $T_i$ has finite variance.  Here is the question: can we prove that $\overline{T}$ is optimal in some way? To me conceptes from MLE estimators or sufficient statistics are a bit difficult to apply, since $E[T]$ is not a parameter of the distribution but maybe I am missing something? Can we "derive" the sample mean estimator to be optimal according to some criterion in the general case and without making assumptions on the underlying distributions ?
While learning about Random variables I came across the mean of random variable X.  The definition says that the expected value of random variable E(X) = Mean of Random variable X  I am not able to understand why is that so. Can any one please help me with it
As the title says, I've heard two possible meanings for turning down the air conditioning: It could mean set the target temperature lower (i.e. colder) or make it work less (i.e. warmer).  Turning down the heat makes it colder, but both meanings are in agreement there. Graphics file extensions and their order of inclusion when not specified When using 's \includegraphics you can specify the file extension or not. For generic use and perhaps portability, extensions are commonly dropped. The types of "approved" inclusions and their order of precedence depends on the driver/compiler being used. What are these driver-defaults?  Related: Visual Studio Editor does not underline errors anymore My Visual Studio (2008) Editor has stopped to underline Errors (this nifty wavy red lines). I can't really tell when, but it can be related to the installation of .Net Framework 3.5 SP 1 or the MVC Beta (which I guess is unlikely). Furthermore have I installed and uninstalled both CodeRush and Resharper for evaluation purposes (decided not to keep either one of them).  Does anyone know the problem and how to restore this functionality again? Find $\lim_{n \to \infty} \left[\frac{(n+1)^{n + 1}}{n^n} - \frac{n^{n}}{(n-1)^{n-1}} \right]$ (a question asked at trivia) My friend's trivia league had this math question:  $$\lim_{n \to \infty} \left[\frac{(n+1)^{n + 1}}{n^n} - \frac{n^{n}}{(n-1)^{n-1}} \right]$$  After computing a few values, one could guess the answer is $e$ = 2.718...But how can we prove that is the limit?  Someone offered up a hand-wavy proof like this:  \begin{align} \lim_{n \to \infty} \left[\frac{(n+1)^{n + 1}}{n^n} - \frac{n^{n}}{(n-1)^{n-1}} \right] &amp; = \lim_{n \to \infty} \left[\frac{(n+1)(n+1)^{n}}{n^n} - \frac{n \cdot n^{n-1}}{(n-1)^{n-1}} \right] \\ &amp;= \lim_{n \to \infty} \left[(n+1)\frac{(n+1)^{n}}{n^n} - n\frac{n^{n-1}}{(n-1)^{n-1}} \right] \\ &amp;= \lim_{n \to \infty} \left[(n+1)\left(1 + \frac{1}{n} \right)^n - n\left(\frac{n - 1 + 1}{n-1} \right)^{{n-1}} \right] \\ &amp;= \lim_{n \to \infty} \left[(n+1)\left(1 + \frac{1}{n} \right)^n - n\left(1 + \frac{1}{n-1} \right)^{n-1} \right] \\ &amp;= \lim_{n \to \infty} \left[(n+1)e - n \cdot e \right] \\ &amp;= \lim_{n \to \infty} e \\ &amp;= e \end{align}  The part about substituting $e$ is hand-wavy since technically this is an indeterminate form of $\infty - \infty$. And using $e$ as as upper bound did not lead me to an easy proof either.  Is there a way to rigorously prove the limit? I tried a few approaches: (a) sandwiching the limit--I could prove $e$ was a lower bound, but I could not find a suitable upper bound converging to $e$, (b) using L'Hopital's rule with no luck, (c) using the mean value theorem--but that also got complicated.  So this is a pretty tough problem to ask at trivia! Is there a way to prove this limit formally?  Sources  Trivia question:   Thread on trivia:   Hand-wavy proof:   Idea for mean value theorem: Why are the longer edges being beveled more? I'll preface this by saying I'm very new to Blender - I'm trying to put a bevel on an edge, looks something like this:    As you can see, the vertical side is much longer than the horizontal side.  When I apply to bevel to the edge there, the bevel stretches proportionally - so there's 'more' bevel on the longer side.  I want to bevel without any 'ratio' like that - just a 45-degree bevel, if that makes any sense.  Do I have to put in a loop cut to make the side lengths equal?  I tried using a sub-surf modifier, but it applies on a bunch of nearby edges, when I just want to round this one corner. Why aren't the braces showing Use of pizza stone in gas oven I have a new gas kitchen stove.  The oven seems to burn so easily. I have a thermometer in there and it is registering the temp that I have the oven set on.  I have seen where people have used a pizza stone to even out the heat.  I want to know if you place a rectangle pizza stone directly on the bottom of the oven or should it be placed on a rack on the lowest setting?  I would appreciate replies from anyone who has used this and if it had helped the problem Any thoughts on a semidecadian badge for people with five years service and an average of N points per year? Given that Stack Overflow is close to its fifth birthday as a public site, has any thought been given to a 'semidecadian' badge for people who have:   been registered with the site five years (a semi-decade, hence the name), scored more than N points per year on average (where N might be 1000).   The idea behind 'average of N points per year' is to allow slow starters to gain the badge eventually, rather than ruling them out in year 1.  The details are obviously subject to tweaking.  This might be a gold badge (the platinum version would be the decadian badge, with a larger value of N required — and no-one eligible for another 5 years or so).  These would complement the current Yearling badge (which is silver). While I basically understand what a determinant is, I wonder how this idea was developed?  What was the principal idea behind its origination?  I would like to know this so that I can have a better conceptualization of the determinant. Use of substitution into the integral $\int_0^2 \frac{\arctan x}{x^2+2x+2}dx$ Why is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ not a UFD? How to access object properties containing special characters? I am currently trying to establish a clear picture of pure/mixed/entangled/separable/superposed states. In the following I will always assume a basis of $|1\rangle$ and $|0\rangle$ for my quantum systems. This is what I have so far:   superposed: A superposition of two states which a system $A$ can occupy, so $\frac{1}{\sqrt{2}}(|0\rangle_A+|1\rangle_A)$ seperable: $|1\rangle_A|0\rangle_B$ A state is called separable, if its an element of the (tensor)product basis of system $A$ and $B$ (for all possible choices of bases) entangled: $\frac{1}{\sqrt{2}}(|0\rangle_A|1\rangle_B+|1\rangle_A|0\rangle_B)$ is not a state within the product basis (again for all possible bases). mixed state: Is a statistical mixture, so for instance $|1\rangle$ with probability $1/2$ and $|0\rangle$ with probability $1/2$ pure state: Not a mixed state, no statistical mixture   I hope that the above examples and classifications are correct. If not it would be great if you could correct me. Or add further cases, if this list is incomplete.   On wikipedia I read about      Another way to say this is that while the von Neumann entropy of the whole state is zero (as it is for any pure state), the entropy of the subsystems is greater than zero.   which is perfectly fine. However I also read on wikipedia a criterion for :     Another, equivalent, criterion is that the von Neumann entropy is 0 for a pure state, and strictly positive for a mixed state.   So does this imply that if I look at the subsystems of an entangled state, that they are in a mixed state? Sounds strange... What would be the statistical mixture in that case?  Moreover I also wanted to ask, whether you had further illustrative examples for the different states I tried to describe above. Or any dangerous cases, where one might think a state of one kind to be the other? I am having trouble adding an image picker to a block form defined in a custom module.  If I create a custom block type I can define an image field as entity_reference and in the block's form view tell it to use entity_browser and media_entity_browser.  This provides me with an picker with which I can select my image:    I am trying to recreate this in a block provided by my custom module, without much success  // /src/Plugin/Block/MyCustomBlock.php  function blockForm($form, FormStateInterface $form_state)  {     ...      $form['image'] = [       '#type' =&gt; 'entity_browser',       '#entity_browser' =&gt; 'media_entity_browser',       '#title' =&gt; $this-&gt;t('Image'),       '#default_value' =&gt; $this-&gt;configuration['image'] ?? null,       '#weight' =&gt; '0',     ];      ... }   This gives me the following.  Note the lack of the bounding box, title and option to edit remove items:     Exporting the configuration files for the working form view gets me this:  // core.entity_form_display.block_content.lead_block.default.yml      ...  field_image:     weight: 1     settings:       entity_browser: media_entity_browser       field_widget_display: rendered_entity       field_widget_edit: true       field_widget_remove: true       selection_mode: selection_append       field_widget_display_settings:         view_mode: default       open: false     third_party_settings: {  }     type: entity_browser_entity_reference     region: content   ...   But if I try and use 'type' =&gt; 'entity_browser_entity_reference' in my blockForm function the field disappears altogether.  Looking into various core files I can see that entity_browser is a @FormElement and entity_browser_entity_reference is a @FieldWidget but I don't really understand the difference.  As you might guess I am fairly new to Drupal and I feel like I am missing something fundamental.  I (think I) understand that custom block types are actually nodes internally and that they are subject to different rules to blocks which I define in my custom modules, but I don't fully grasp what the differences here are.  Is is it possible to use the entity browser in my custom module block form in the same way as it is used in the form which I build through the admin's custom block type builder?  Thank you. bash fork: retry: Resource temporarily unavailable I am trying to run a shell script which will create process using a shell script. I get Resource temporarily unavailable error. how to identify which limit (memory/process/filecount) is creating this problem. Below is my ulimit -a results.  core file size          (blocks, -c) 0 data seg size           (kbytes, -d) unlimited scheduling priority             (-e) 0 file size               (blocks, -f) unlimited pending signals                 (-i) 563959 max locked memory       (kbytes, -l) unlimited max memory size         (kbytes, -m) unlimited open files                      (-n) 65535 pipe size            (512 bytes, -p) 8 POSIX message queues     (bytes, -q) 819200 real-time priority              (-r) 0 stack size              (kbytes, -s) unlimited cpu time               (seconds, -t) unlimited max user processes              (-u) 10000000 virtual memory          (kbytes, -v) unlimited file locks                      (-x) unlimited What are tag synonyms? How can tag synonyms be distinguished from "real" tags? How are new tag synonyms created? Who can create tag synonyms? How can I delete/reverse/undo bad tag synonyms? What are merged tags? Multiplicative inverse questions This is my first time posting on Math exchange. I have been self-teaching myself Mathematics and recently started learning some Algebra. I was posed the following questions, in which I would like to answer before I proceed, but having some trouble finding proper solutions. So the format will be question, and my answer.      What name do we give an element $a \in \mathbb{Z}_n$ that has a multiplicative inverse?   A modular multiplicative inverse of an integer $a$ is an integer $x$ such that $ax \cong 1 (\mod m)$. This means that the remainder after dividing $ax$ by integer $m$ is equal to $1$.      Give an example, for some $n \in \mathbb{N}$, of an element of $\mathbb{Z}_n$, that does not have a multiplicative inverse?   This question is throwing me off because - does the $n$ we speak of matter for both $n \in \mathbb{N}$ and $\mathbb{Z}_n$ ot solely $n$? My guess here would be $0$ because we cannot have an inverse of $0$.      Prove that multiplicative inverse are unique; an element $a \in \mathbb{Z}_n$ cannot have two multiplicative inverses.    Suppose that an element $a\in\mathbb{Z}_n$ has an inverse $a'$. Next, suppose that $a^*$ is also an inverse of $a$. Then, $a'$ is a solution to $a \cdot_n x=1$, and similarly $a^*$ has a solution to $a \cdot_n x=1$.   Lemma: Suppose $a'$ is multiplicative inverse of $a$ in $\mathbb{Z}_n$, then for any $b \in \mathbb{Z}_n$ the equation $a \cdot_n x=b$ has a unique solution, such that $x = a' \cdot_n b$.   By Lemma, equation $a \cdot_n x = 1$ has a unique solution, namely $x=a'=a^*$. Therefore, an element $a\in \mathbb{Z}_n$ cannot have two multiplicative inverses.    Thank you for the help. I would love any constructive feedback for these. If you can help me elaborate on some ideas, that would be really helpful also.  Thank you very much! How can I install a Canon printer or scanner driver? Only charge capacitor when button pushed then turn on LED momentarily with capacitor when button released I am trying to make circuit to turn on an LED momentarily. I would like to charge the capacitor when the button is pushed, then when released turn on the LED momentarily with the capacitor. Is the High Elf's extra cantrip in addition to cantrips from your class?