Define $Var(XY)$ in terms of $E(X)$, $E(Y)$, $Var(X)$, $Var(Y)$ for Independent Random Variables $X$ and $Y$
Variance of product of multiple independent random variables
When should we use proximity rule in "either/or", and "neither/nor"? According to this , if at least one of the nouns involved is plural then it should take the plural form of the verb. Otherwise, it should take the singular form of the verb. But in the last part it says not all grammars agree to those rules thus proximity rule applies. However, it doesn't explain when to apply the proximity rule and when not to use the former rules. How to prove $x^a-1\mid x^b-1 \iff a \mid b$, where $x \ge 2$ and $a,b,x \in \Bbb Z$.  I've tried the following in attempting to solve this:  $$a\mid b \Rightarrow aq=b \Rightarrow x^{aq}=x^b \Rightarrow x^ax^q=x^b$$  Because $x^q \in \Bbb Z$, it follows that $x^a\mid x^b$.  This is as far as I have gotten; any help getting further is appreciated.  Note: It may be that this identity is not true at all? 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 Alignment issue in Area 51 main site - site switcher, inbox and achievements In Cycles, is there a way to create a procedural texture that follows the longitudinal lines of a sphere? Like a beach ball or globe? The texture should be able to be mapped to any sphere-shaped object, not just perfect spheres.    My current approach is to use a rotated wave texture, but that doesn't come from the poles (It uses latitudes) When did 5 foot squares become standard in D&D? Method defined as testMethod do not support web service callouts, test skipped This is a batch file in Windows.  Here is my .bat file          @echo off copy "C:\Remoting.config-Training" "C:\Remoting.config"  "C:\ThirdParty.exe"   This works fine except the .bat file leaves the command window open the whole time the "ThirdParty" application is running. I need the command window to close.   I would use the short-cut for the application but I must be able to run this copy command first (it actually changes which data base and server to use for the application).    The ThirdParty application does not allow the user to change the source of the db or the application server.    We're doing this to allow users to change from a test environment to the production environment. Flag abusive users I think the community does a great job policing questions/answers, etc. and the community has some good tools available to do this.  Between flagging questions with some reasons behind it and editing posts, closing them, and deleting them, the only thing that seems to be missing to me is the ability to flag a user.  Reasons to flag a user as abusive:   Duplicate User Abusive Behavior Offensive user name or profile User profile consists of spam Others? raw_input in python without pressing enter When the loop is the figure shown below will be rotated in clockwise direction the the slip ring connected to the red segment of the loop will give us the positive voltage and the slip ring connected to the black segment of the loop will give us negative voltage according to the Flemings right hand rule.  What if the loop is rotated in counter clockwise direction ? Will the slip rings provide us the same polarities which they were giving when the loop was moving clockwise ?  Plus  The voltage from the loop is given by the equation.  V(t) = Vmaximum * sin (theta)  Where theta is the positive angle the red segment makes in the clockwise direction or in counter clockwise direction from this position ? Are two norms equivalent if they induce the same topology on a vector space? What is the relationship between size of object with distance? How does the size/length of an object vary with distance?   Is it a logarithmic relationship? exponential? linear?   I plotted a curve of the size/length of an object for different distances from the camera, and the curve looked exponential/logarithmic. I was trying to understand the reasoning behind that. $n$-th digit in the sequence of natural numbers Can the sun damage the camera sensor? Under what conditions? I want to experiment taking photos in which the sun appears. I'm afraid of what might happen if I take one with a narrower angle (where sun would be bigger). Can the lens act as a magnifying glass and burn the CCD or CMOS sensor?  Under which circumstances (zoom, exposure, aperture, etc...), can the sensor be damaged by the sun? How do you report percentage accuracy for glmnet logistic regression? Footnote to an item in a description environment It seems like it's not possible to put footnote inside a description list item.  For example in the following code first footnote appears at the bottom of the page, but the second - doesn't:  \documentclass[a4paper,12pt]{article}  \begin{document}     Text text text.\footnote{This footnote is working}     \begin{description}         \item[Label\footnote{This footnote is not working}] Some description     \end{description} \end{document}   Why it's not working this way? And is there any way to put footnote to a list item? How do you challenge the closing of one of your questions? Is there a way to make IE8 honour opacity on an `:before` pseudo element? What is this sand covered insect? I have found in a wooden house, in the mountains of central Spain this very curious insect:    I was wondering if that was its skin or whether it's decomposition or sand or saw-dust.  That particular region of Spain does have stink-bugs ... but this one doesn't look like anything I've seen before.  A google image search returns mostly arachnids ... but it seems to have only 6 legs.