We are given an i.i.d sample of size n, $X_{1},...,X_{2}$.  We also know that for all i, $EX_{i}=\mu,Var{X_{i}}=\sigma^{2}$.   Let us define $\hat{X}_{median}$ as the median of these samples:  $\hat{X}_{median}(X_{1},...,X_{n})=median(X_{1},...,X_{n})$ .   Is there any known way to explicitly express the mean and variance of the median estimator ($E(\hat{X}_{median}), Var(\hat{X}_{median})$) In terms of $\mu,\sigma$?  Thank you!
Given a sample of n units out of a population of N, population median can be estimated by the sample median.  How can we get the variance of this estimator?
SEO repeated keywords in domain and url Why are my photos not crisp? I use a Canon EOS Rebel T3i (600D) with the kit lens. I have experimented a lot with different settings and modes, but I feel like I have wasted many good shots that could have looked amazing, if only they were a little bit sharper.  I use auto-focus on most of my pictures and make sure to choose the right focus point, but all of my photos are still a little bit out of focus. When I zoom in, I can see that the pixels are not very crisp or well-aligned.  I am not sure if this issue is caused by my equipment or skills. Putting char into a java string for each N characters I have a java string, which has a variable length.   I need to put the piece "&lt;br&gt;" into the string, say each 10 characters.   For example this is my string:  `this is my string which I need to modify...I love stackoverlow:)`   How can I obtain this string?:  `this is my&lt;br&gt; string wh&lt;br&gt;ich I nee&lt;br&gt;d to modif&lt;br&gt;y...I love&lt;br&gt; stackover&lt;br&gt;flow:)`   Thanks A dollar amount $\${-}67.9{-}$ is divisible by 72. What is this number? A client buys 72 turkeys. On the receipt, 2 digits are missing, the first and the last. So, the total price is $\${-}67.9{-}$. Find a way to make the price of each turkey round at the second digit without trial-and-error method.  So, the answer is obviously $\$5.11$/turkey, so $\$367.92$ total. I found that with with 2-3 trial-and-error.  I bypassed his problem by using python and made a script that tries every combination and assures it is divisible by 0.01.   Please help me find a mathematical way of solving this problem. Why is the sum of the digits in a multiple of 9 also a multiple of 9? The sum of the digits in $9 k$ (where $k$ is an integer) is a multiple of $9$: for example   $$9\cdot 1=9$$ $$9\cdot 7=63 \qquad  \text{and } 6+3=9\cdot 1$$ $$9\cdot 11=99 \qquad \text{and }  9+9=9\cdot 2$$  But why? Prove the inequality and limitation Suppose $\{a_k\}$ ,$\{b_k\}$ and $\{\xi_k\}$ are non negative, and for all $k\ge 0$ , we have $$a_{k+1}^2\le(a_k+b_k)^2-\xi_k^2$$   Prove   1.$$\sum_{i=1}^{k}\xi_i^2\le(a_1+\sum_{i=0}^kb_i)^2$$   2. If $\{b_k\}$ also satisfy $\sum_{k=0}^{\infty}b_k^2\lt+\infty $, then $$\lim_{k\to\infty}\frac{1}{k}\sum_{i=1}^k\xi_i^2=0$$   After doing this transformation:$$\xi_k^2\le(a_k+b_k)^2-a_{k+1}^2$$ $\Rightarrow$ $$\sum_{i=1}^{k}\xi_k^2\le(a_1+b_1-a_2)(a_1+b_1+a_2)+(a_2+b_2-a_3)(a_2+b_2+a_3)+\cdots(a_k+b_k-a_{k+1})(a_k+b_k+a_{k+1})$$ then I can't move forward any more. Some time ago I closed my front door (rather forcefully) while the deadbolt was sticking out partially.  This loosened the deadbolt, and the top screw no longer held. In addition, it cracked the door on the interior side. I've attached a picture showing the damage.   The door works mainly, though it can be difficult to lock sometimes, especially when the weather gets colder.  I'm currently renting, and will soon be moving out of this house, once I buy a house. I would like to repair this door myself, and I'm hoping I don't have to replace the entire door.  Is this something that can be repaired? If so, how? I realize that a new deadbolt is probably needed, but how can I repair the wood damage around it? How can I find my hardware details? Is there any built-in software or terminal method allowing me to view the hardware profiles on my system? Windows equivalent of such a feature would be Device Manager. Solving $\sin x\cos3x+\cos x\sin3x=\sqrt{3}/2$ in the interval $[0,2\pi]$ Solve the following trig equation: $$\sin(x)\cos(3x)+\cos(x)\sin(3x)=\frac{\sqrt{3}}{2}$$ in the interval $[0,2\pi]$.  Using trig identities, it is now at $$\sin(x+3x)=\sqrt{3}/2 \tag{1}$$ Next, it has become $$\sin(x+3x)=\sin 60^\circ \tag{2}$$ leading to $$\sin(x+3x)-\sin60^\circ=0 \tag{3}$$ From there, it is unclear what to do. Is there a way to transfer money and items from iOS to Android? I've been playing Jetpack Joyride on iOS, and I just got an Android device. Can I move my things, such as money and unlocks, from my iOS device to Android? Active vs passive voice with inanimate object as the subject I have asked the  in ELL Stackexchange forum. There all the answers and comments I received was really helpful, but still I was looking for something more. So I have decided to ask this same question here also.   I have seen some sentences like this -         The film was announced to be released on this coming Friday.   The film was announced to release on this coming Friday.      Both the sentences are used, and I know they are interchangeable and does bear the same meaning. But I have some doubt about the second sentence - why is it "to release", when the film cant release itself.   Some other sentences of similar structure that arises my confusion -         The product sells well.   The book reads well.   The glass broke.      The sentences are understandable, but in each of the sentences the subjects can't perform any job, as they all are inanimate object. The verbs are active, but they acts like they are used as passive.  Now my question is how to decide which verb to use this way and which not? After upgrading to macOS Sierra, I'm getting the notification that "Your disk is almost full. Save space by optimizing storage.":    The options appear to be to store my files in iCloud, automatically delete files, or manually delete files:    My problem/irritation is that I have 80GB free of my 440GB volume.  The question: is there a way to silence this notification in a (semi-)permanent fashion, or to change the threshold it uses for the notification? Which is more correct:     Apropos of your earlier comment, I decided to....   or      Apropos your earlier comment, I decided to...   Actually, apropos is so fancy a word, even I, a word maven if I do say so myself, am not entirely sure when to use it.  If I may add to my preposition question, are there any constructions where it would be appropriate to use something like with regard to and not apropos? 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! wilcox.test usage in R I would like to know if there is a difference in the values taken from measurements of a given characteristic sampled at primary forest (Point=1) and secondary forest (Point=0)  I'm using two vectors, one with the data itself (DATA$Measurements), and the other one with the category of sampling point (DATA$Point).  How should I perform the wilcox.test fucntion?  I'm doing this way:  wilcox.test(DATA$Measurements~DATA$Point) But I notice that are other forms of performing this test:  wilcox.test(DATA$Measurements,DATA$Point) wilcox.test(DATA$Measurements,DATA$Point,paired=TRUE) Which one is more suited for testing my question of interest (Is there a difference between points for this given characteristic), and what is the difference between these options?? I'm a little confused as to when to use  for my physics class. For example, I'm asked to find the average speed of a race car that travels around a circular track with a radius of $500~\mathrm{m}$ in $50~\mathrm{s}$.  Would I need to apply the rules of significant figures to this step of the problem? $$ C = 2\pi (1000~\mathrm{m}) = 6283.19 $$  Or do I just need to apply significant figures at this step? $$ \text{Average speed} = \frac{6283.19~\mathrm{m}}{50~\mathrm{s}} = 125.664~\mathrm{m}/\mathrm{s} $$  Should I round $125.664~\mathrm{m}/\mathrm{s}$  to  $130~\mathrm{m}/\mathrm{s}$ since the number with the least amount of significant figures is two? I am trying to figure out how to add a Lightning Web Component to a Lightning tab.  I have it on the Home Page, but I would like it to be on its own tab.  I feel like there should be an option to expose it to the Lightning tabs.  When I am in the Tabs section, I don't see my LWC listed. Find Java classes implementing an interface Some time ago, I came across a piece of code, that used some piece of standard Java functionality to locate the classes that implemented a given interface. I know the functions were hidden in some non-logical place, but they could be used for other classes as the package name implied. Back then I did not need it, so I forgot about it, but now I do, and I can't seem to find the functions again. Where can these functions be found?  Edit: I'm not looking for any IDE functions or anything, but rather something that can be executed within the Java application. Pure significance of line integrals of vector fields I can understand how the line integral of a scalar function has pure-mathematical significance, representing an area beneath a curve. But is there a pure significance to the line integral of a vector field? The only understanding I have of this is the physical example of work done on a moving particle. Thanks for any insight. It's not clear to me how to choose Spells Known when creating a new character at a higher level.  Let's take the Bard as a working example first:  Bard level 2:   Spells Known: 5 Slots per Spell level: 1st &times; 3   Bard level 3:   Spells Known: 6 Slots per Spell level: 1st &times; 4, 2nd &times; 2   I'm interpreting this to mean that, if an existing 2nd-level Bard gains a new level (level 3), he's gaining 1 (one) new spell known. He can then choose the new spell known to be either 1st or 2nd, correct?  What I don't understand then, is the case of creating a new character starting at a higher level; say, creating a new Bard at level 3.  In that case you're forced to choose 6 spells, which must be from the list of 1st- and 2nd-level spells, correct?  When making this 3rd-level Bard from scratch, though, can I select 2nd level spells for all 6 spells known?  As another example, say you're creating a very high level Bard from scratch, starting at level 15. The book says your Spells Known is 19, and your spell slots are 1st × 4, 2nd × 3, 3rd × 3, 4th × 3, 5th × 2, 6th × 1, 7th × 1, and 8th × 1.  So, for those 19 Spells Known, you can choose any spells from 1st through 8th spell level. Am I free then to select as many spells as I can at 8th spell level (5 in number), working backwards, and in the end ignoring the 1st, 2nd, 3rd level spells entirely?  As another example, the Ranger doesn't know any spells at 1st level, nor have any spell slots, but the book says “You know two 1st-level spells of your choice from the ranger spell list.” Does it mean at 2nd level and above?  Long story short: When choosing Spells Known for a spellcasting character created above 1st level, am I allowed to choose my Spells Known at whatever spell level I like, just so long as I have at least one spell slot of that level?  I'm working on a D&amp;D 5e character generator and need to clarify this.