Let $X \sim exp(\lambda)$ be independent of $Y \sim exp(\lambda)$ compute $E[X|X+Y]$
Compute $E(X\mid X+Y)$ for independent random variables $X$ and $Y$ with standard exponential distributions
If $X$ is an orthogonal matrix, why does $X^TX = I$? It's not immediately clear to me why this is true. My notes say that putting $n$ orthonormal vectors $ v_1, ..., v_n$ in the columns of $X$ gives $X^TX = I$, and it follows from this that the rows of $X$ are orthonormal.   Can you explain this better to me? I tried the $2\times 2$ case, but it isn't clear to me why this is true. Why doesn't matter pass through other matter if atoms are 99.999% empty space? The ghostly passage of one body through another is obviously out of the question if the continuum assumption were valid, but we know that at the micro, nano, pico levels (and beyond) this is not even remotely the case. My understanding is that the volume of the average atom actually occupied by matter is a vanishingly small fraction of the atom's volume as a whole. If this is the case, why can't matter simply pass through other matter? Are the atom's electrons so nearly omnipresent that they can simultaneously prevent collisions/intersections from all possible directions? Writing comments in QGIS Graphical Modeler I want to explain my model step by step so anyone can easely understand what's going on. So, is it possible to write comments in the QGIS's Graphical Modeler? How to make a meta box field a requirement Importance of PTR records on nameserver IP's? I have not yet experienced any DNS failures due to a missing PTR record, but am wondering:   Is there any added value in having a valid PTR record on a nameserver IP-address? Transpose/Unzip Function (inverse of zip)? Installing two glibc alongside in debian/ubuntu Is it possible to install and use two different glibc versions on the same machine. Where one version is only used to run legacy software, which relies on old glibc binaries?  Is it possible to do that with the aid of the package manager (something like "install this package, and its dependencies, to /opt/old-glibc), instead of managing it dire Detecting significant predictors out of many independent variables In a dataset of two non-overlapping populations (patients &amp; healthy, total $n=60$) I would like to find (out of $300$ independent variables) significant predictors for a continuous dependent variable. Correlation between predictors is present. I am interested in finding out if any of the predictors are related to the dependent variable "in reality" (rather than predicting the dependent variable as exactly as possible). As I got overwhelmed with the numerous possible approaches, I would like to ask for which approach is most recommended.    From my understanding stepwise inclusion or exclusion of predictors is  E.g. run a linear regression separately for every predictor and correct p-values for multiple comparison using FDR (probably very conservative?) Principal-component regression: difficult to interpret as I won't be able to tell about the predictive power of individual predictors but only about the components.  any other suggestions? Difference between 'struct' and 'typedef struct' in C++? Number of possible permutations of n1 1's, n2 2's, n3 3's, n4 4's such that no two adjacent elements are same? What is the difference between lands and mana? Can I fetch the value of a non-standard CSS property via Javascript? I am trying to read a custom (non-standard) CSS property, set in a stylesheet (not the inline style attribute) and get its value. Take this CSS for example:  #someElement {   foo: 'bar'; }   I have managed to get its value with the currentStyle property in IE7:  var element = document.getElementById('someElement'); var val = element.currentStyle.foo;   But currentStyle is MS-specific. So I tried getComputedStyle() in Firefox 3 and Safari 3:  var val = getComputedStyle(element,null).foo;   ...and it returns undefined. Does anyone know a cross-browser way of retreiving a custom CSS property value?  (As you might have noticed, this isn't valid CSS. But it should work as long as the value follows the correct syntax. A better property name would be "-myNameSpace-foo" or something.) Short Story about interminable rain on Venus It's not All Summer in a Day.  This was a story about explorers on Venus, who are separated from their base (possibly they're lost, possibly they suffered equipment failure, possibly they're just traveling from base to base) and are slowly being driven insane because it never stops raining.  The story is old enough that Venus' clouds were still believed to indicate a wet climate.  I can't remember much more detail than that.  I have the impression that the narrator at one point either dreamed of or fantasized about walking into the destination base and being handed a very plush towel to dry off.  I can't remember the central plot points - only the (very wet) setting.  I would have read this in the late 1970's or early 1980's.  I thought it might be an H. Beam Piper short story but I've been through his catalog and don't see any likely suspects. Duplicate questions on other SE sites? I recently asked a  on security.stackexchange.com. The question was related to XSS so in my opinion it belonged to the security website.  There was a comment on the question saying that a similar question exists on SO and my question was a duplicate (I think he gave it a down vote). The guy was right but the question was asked on another website (SO). There may be a case when person asking a question might not have an account with the other website.  Is it a good practice to mark questions duplicate if you have seen a similar question on some other website? Is acceleration due to gravity constant? URL regex passed by regex buddy but failed by Dart A Light is shone out the side window of a spaceship ( moving close to light speed ), will the light still travel at the speed of light in the B axis? I've seen many posts regarding a combinatorial proof of the following question. But for a non-combinatorial proof would the following method work? Also... is this the easiest way to arrive at a proof? It seems to be rather verbose.  Show the formula $k {n\choose k} = n {n-1 \choose k-1}$ is true for all integers $n,k$ with $0\le k \le n$.  My answer:  Observe that  $$\begin{align*} {n\choose k}&amp;= \frac {n}{k} {n-1 \choose k-1}\\ &amp;= \frac {n}{k}\left[ {n-2 \choose k-2} + {n-2 \choose k-1}\right]\\ &amp;= \frac {n}{k}\left[ \frac{(n-2)!}{(k-2)!(n-2-(k-2))!} + \frac {(n-2)!}{(k-1)!(n-2-(k-1))!}\right]\\ &amp;= \frac {n}{k}\left[ \frac{(n-2)!}{(k-2)!(n-k)!} + \frac {(n-2)!}{(k-1)!(n-k-1)!}\right]\\ &amp;= \frac{(n-1)!}{(k-1)!(n-k)!} + \frac {(n-1)!}{(k)!(n-k-1)!}\\ &amp;= \frac{(n-1)!}{(k-1)!(n-1-(k-1))!} + \frac {(n-1)!}{(k)!(n-1-k)!}\\ &amp;={n-1 \choose k-1} + {n-1 \choose k}={n \choose k}. \end{align*}$$ So I have a Palm Pre (original P100EWW) model that I enabled developer mode on, and installed a Debian Squeeze chroot.  Works great.  I have plans to use this for ANYTHING (bittorrent peer, web server) but a phone.  I noticed if I do a cat /dev/urandom &gt; /dev/fb0 it actually writes random pixels to the screen until a No space left on device error is generated.  Awesome, now I can use the display.  So what kind of utilites are there that will either A) let me use /dev/fb0 as a console I can output text to or B) render text on /dev/fb0 from the command line.  I don't know about recompiling the kernel for this yet (I'd love to eventually strip WebOS off entirely and turn this into a minimal ARM server) so userspace tools if they exist is what I'm asking about.  Also would prefer to render directly to /dev/fb0 and not use X. So I'm trying to prove the following two inequalities: $$\frac{\sqrt{n}}{2} \leq \phi(n) \leq n.$$ The upper bound we get from simply noting that $\phi(n) = n \prod_{p | n}\left( 1 - \frac{1}{p}\right)$ and the fact that $(1 - \frac{1}{p}) \leq 1$. But how can we get the lower bound? I tried using the same expression for $\phi(n)$ but it seems to not really give me the inequality. Can you help?