If  $X$ and $Y$ are independent random variables, then can I say that $X^2$ and $Y$ are independent?
When proving $\bar x$ and $S^2$ are independent in my noted it says that "functions of independent quantities are independent ". Can someone tell me how functions of independent quantities are independent happen?  Also let $X_1,X_2,X_3,\dots,X_n$ be a random sample.And suppose we want to estimate parameter $\theta$ using $T(x)$ as an estimator. In this I have a expression as $E\{ [  T(X)-E(T(X))][E(T(x)-\theta)]\}$.  I want to know if I can say that in $[T(X)-E(T(X))]$ since $E(T(X)$ (expected value is a one particular fixed value) and $T(X)$ are independent of one another. Also the expression $[E(T(x)-\theta)]$ is independent of $X$.  Hence the two expressions $[T(X)-E(T(X))]$ and $[E(T(x)-\theta)]$  are independent of one another. So that I can use if $a,b$ are independent $E(ab)=E(a)E(b)$ form
Restore Macintosh HD to its original partition configuration Let $(e_n)$ (where $ e_n $ has a 1 in the $n$-th place and zeros otherwise) be unit  standard vectors of $\ell_\infty$.   Why is $(e_n)$ not a basis for $\ell_\infty$?  Thanks. latex : Exclude draftwatermark from specific pages I am using draftwatermark package. Is there any option to exclude watermark from some specific pages? Say no watermark from pg 3-4. I can not use xwatermark package. It conflicts with the geometry package of my document.  MWE:  \documentclass[12pt,a4paper]{exam} \usepackage{draftwatermark}  \usepackage{lipsum}   \SetWatermarkAngle{45}  \SetWatermarkLightness{.8}  \SetWatermarkFontSize{10cm}  \SetWatermarkScale{5}  \SetWatermarkText{\tt{12345}}  \begin{document}  \lipsum[1-20]  \end{document} About the structure of ''There are/is'' I want to discuss the structure of ''There are/is''. First of all, what is the difference between:  There are three children.  and  Three children exist.  Do they have the same meaning? And secondly, In my native language, we would say that  Üç çocuk vardır.  &quot;Üç çocuk&quot; = Three children, &quot;var&quot; = existence (something  that exists, a being), and &quot;dır&quot; is a copula. So basically, this structure is not different from -for example- ''three children is a car''. ''a car'' and ''existence'' have the same function. But ın English, that structure that I have mentioned is confusing for me. I know that you are all used to it. Some can say ''What's the problem with that? It is just ''there are'' and ''there is.'' ''. I realize that I still did not ask a question. Well, let me try to ask a good one, I guess. Normally, ''there'' means a place where I am not in that place. Is there any relevance between ''there'' that means a place where I am not in that place AND ''There'' that is used in the structure of ''There are/is''? What kind of structure is this? By the way, I am new in English. Please excuse my naive usage of language. What do you think about it? horizontal alignment of dot over hat over bold greek letter! For some strange reason, this:   \hat{\dot{\bm{\phi}}}   prints the dot and the hat slightly off the left of the phi letter. Any ideas if this can be fixed? thanks. Using bm package for bold symbols. Caption issue when placing figure side by side I placed 2 figures side by side using the minipage method from , and the figures look perfectly fine. However, the caption of the right figure stick out into the margin, why is that and how can I fix it?    My code is like this:  \begin{figure}[htbp]   \centering   \begin{minipage}{.5\textwidth}     \centering     \includegraphics[width=6.6cm]{figure1.png}     \caption{Caption for figure 1.}     \label{fig:fig1}   \end{minipage}%   \begin{minipage}{.5\textwidth}     \centering     \includegraphics[width=6.6cm]{figure2.png}     \caption{Caption for figure 2, but longer.}     \label{fig:fig2}   \end{minipage} \end{figure} How to set default magnification for all windows Java Round up Any Number First time player in a group figuring out things as we all go.  I just hit level four as a Ranger, and I get to improve one ability score of my choice by 2, or I can increase two ability scores of my choice by 1.  My first thought was Strength, as I've been rocking a score of 11 with a mod of +0 so far. But I read that Rangers use (wisdom mod + 10) for their spell casting modifier. Currently my wisdom is 13, with a mod of +1. If I increase either STR or WIS, will the modifier increase as well? How and why?  A Monk in our party went from 9 CON to 11 during their ability score improvement, and by doing so their modifier went from −1 to 0.  We're all pretty new, so between eight people we're currently sharing one book, and all trying to locate our own copies for later campaigns. That's making it… inefficient for us to learn these things quickly. If G is not commutative, then is there always a subgroup that is not a normal subgroup? Interview Question Asked In yahoo Can you find the smallest positive number such that if you shuffle the digits of the number in a particular order, the shuffled number becomes twice the original number.  Source:   I understand the answer is $125874 =&gt; 251748$  $251748$ is twice the $125874$ and have same digits $1,2,4,5,7$ &amp; $8$  but how to solve this non programmatic ? My Ubuntu 12.04 was normally updated through the Update Manager, something about gtk. Then I rebooted and now all that shows after the splash screen is a blue screen that only shows the cursor. Confused about complex numbers I am confused about something: \begin{eqnarray}     (e^{2 i \pi})^{0.5} = (e^{2 i \pi \cdot 0.5})= e^{i \pi}=-1 \end{eqnarray} but \begin{eqnarray}      e^{2 i \pi}=1~ and~ 1^{0.5}=1  \end{eqnarray} where is my mistake??  thanks URLs are linked despite being inside a code span This issue came up when I got different records counts for what I thought were identical queries one using a not in where constraint and the other a left join. The table in the not in constraint had one null value (bad data) which caused that query to return a count of 0 records. I sort of understand why but I could use some help fully grasping the concept.  To state it simply, why does query A return a result but B doesn't?  A: select 'true' where 3 in (1, 2, 3, null) B: select 'true' where 3 not in (1, 2, null)   This was on SQL Server 2005. I also found that calling set ansi_nulls off causes B to return a result. How to download Bootcamp drivers without Bootcamp assistant? I'm setting up my MacBook Air for a Windows-only boot setup. This was easy enough. It already had rEFIt on it, I just connected a Windows 7 boot thumb drive, and away I went.  However, it seems Apple does not let you download Bootcamp drivers outside of their assistant tool. Reinstalling OS X just for the drivers is ridiculous.  This laptop no longer has OS X on it at all. How do I get the required drivers? I cannot find them anywhere on Apple's support pages.  Every other manufacturer has their drivers available for download.   How do I download the Windows drivers for my 2010 MacBook Air without the BootCamp Assistant? I am a Chinese citizen married to a US citizen last year. I am going to apply for a UK tourist visa. The name shows on my passport is my maiden name, while any other documents the UK visa needed, such as bank statement, is with my married name. I can provide my marriage license to prove the link between my maiden name and married mane.   Can I apply for this UK visa with my situation? If asking a question or giving instruction towards a group of objects where the number is unknown (and possibly could be one) what is the correct way to phrase the sentence? Which of the following sentences are more correct:     Has there been any studies regarding this and if so what has been the   conclusion?   OR     Has there been any studies regarding this and if so what have been the   conclusions? We know that if a finite field $F$ has characteristic $p$ (prime), then $F$ has cardinality $p^r$ where $r = [F:\mathbb{F}_p]$.   I'm now trying to say something about the possible cardinalities of subfields of $F$. I can see that there is a subfield of cardinality $p^s$ for each $s$ that divides $r$, given by the fixed field of the group generated by $\phi^s$, where $\phi$ is the Frobenius automorphism.  Now suppose $K$ is a subfield of $F$. Then (since both are additive groups), Lagrange gives us that $|K|$ divides $|F|$, so $|K| = p^t$ for some $1 \leq t \leq r $ (alternatively, $K$ contains $\mathbb{F}_p$ and so is a vector space over $\mathbb{F}_p$ and is thus isomorphic to $\mathbb{F}_p^t$, where $t = [K:\mathbb{F}_p]$). By considering the multiplicative group of units of $K$ and $F$ respectively, we get that $ p^t - 1$ divides $p^r -1$. I want to make the leap to $t|r$, but I'm failing to see why this needs to be true. Any help would be appreciated. Force "low power mode" on mac