$P(X_1 > 0 \mid X_1 + X_2 > 0)$ for IID $X_1, X_2 \sim \mathcal{N}(0,1)$
What is $P(X > 0 \mid X + Y > 0)$ given that $X, Y$ are i.i.d standard normal?
Problem of unbalanced data Should I use the past tense with did? Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ I can't crack this one.      Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ (the only constraint is that  $a,b,c,z \in \mathbb{Z}$ and $c\neq 0)$ I have a script which calls two commands:  long_running_command | print_progress   The long_running_command prints progress but I'm unhappy with it. I'm using print_progress to make it nicer (namely, I print the progress in a single line).  The problem: Connection a pipe to stdout also activates a 4K buffer, so the nice print program gets nothing ... nothing ... nothing ... a whole lot ... :)  How can I disable the 4K buffer for the long_running_command (no, I do not have the source)? Do I need Transit Visa to visit London for 4-5 hours during layover? I get this error: dpkg: unrecoverable fatal error, aborting: files list file for package `fonts-tlwg-kinnari' contains empty filename How do I compare binary files in Linux? I need to compare two binary files and get the output in the form:  &lt;fileoffset-hex> &lt;file1-byte-hex> &lt;file2-byte-hex>  for every different byte. So if file1.bin is    00 90 00 11   in binary form and file2.bin is    00 91 00 10   I want to get something like    00000001 90 91   00000003 11 10   Is there a way to do this in Linux? I know about cmp -l but it uses a decimal system for offsets and octal for bytes which I would like to avoid. What is the definition of a symmetric distribution? What's the definition of a symmetric distribution? Someone told me that a random variable $X$ came from a symmetric distribution if and only if $X$ and $-X$ has the same distribution. But I think this definition is partly true. Because I can present a counterexample $X\sim N(\mu,\sigma^{2})$ and $\mu\neq0$.  Obviously, it has a symmetric distribution, but $X$ and $-X$ have different distribution! Am I right? Do you guys ever think about this question? What's the exact definition of symmetric distribution? The coefficient of kinetic friction can be calculated using $\mu_k = F_k/F_n$. A change in angle does not affect the normal force, but doesn't a change in angle affect the friction? As in if the angle is steeper, wouldn't the object have a greater propensity to move down the slope, thus there will be less friction? It's clear to me by inspection that if we fix $\beta = \frac{1-\mu}{\mu} \alpha$ (thereby fixing the mean) and let $\alpha \rightarrow 0$, the Beta distribution approaches a Bernoulli($\mu$) distribution.  For example:  par(mfrow = c(1, 2),     oma = c(0, 0, 1.5, 0)) xx = seq(0, 1, length.out = 1000) mus = c(.2, .7) for (ii in 1:2) {   mu = mus[ii]   matplot(xx, sapply(10^(-1:-5), function(al)      pbeta(xx, al, (1-mu)/mu * al)),     type = 'l', lty = ii,     main = paste('Mean:', mu),     ylab = 'Cumulative Probability', xlab = 'x') } title('Beta Approaches Bernoulli', outer = TRUE)     That is, if $X \sim B(\alpha, \frac{1-\mu}{\mu} \alpha)$, its CDF $F_X(x; \alpha, \mu)$, satisfies  $$ \lim_{\alpha \rightarrow 0} F_X(x; \alpha, \mu) = 1 - \mu = F_Y(x) \quad \forall x \in (0, 1)$$  Where $Y \sim$ Bernoulli($\mu$), and the convergence is non-uniform. I tried to do a more formal proof of this but couldn't make any progress, even in simple cases (like $\mu = \frac12$). The  makes reference to the $\mu = \frac12$ case several times without proof. Perhaps I'm missing something easy about doing the integral, or some fact about the Beta/incomplete Beta function. Any help would be appreciated! Resistors in series vs parallel When to use 'which' or 'in which' or 'that' (as relative pronouns)? How can I left-align a caption? Generate all combinations from multiple lists Need to prove that $(S,\cdot)$ defined by the binary operation $a\cdot b = a+b+ab$ is an abelian group on $S = \Bbb R \setminus \{-1\}$. Where does the word "dog" come from? Probability to obtain more than X with 3 dice. What happens if I buy a game I already own through Steam a second time? I am debating whether or not to get the Potato Sack (yes, I realize I've been debating for several weeks now) bundle through steam.  I already own: Defense Grid: The Awakening, Super Meat Boy, and Audiosurf, all of which are part of the sack pack. What happens if I buy the Potato Sack?  Do I get 2nd copies of the three above mentioned games to gift elsewhere?  Is the price of the bundle reduced by the games I already have?  Do I get nothing special for already owning those 3 games? Is voting to balance in the spirit of the site? The other day I saw an answer which was ok. Not great but not bad. So no reason to upvote or downvote. However the answer already had a single downvote. I felt this was unfair so upvoted the answer to keep it at what I felt was a fair score. It wasn't until afterward that it occurred to me that I should have perhaps solely voted on the content of the answer.  Is this a reasonable reason for an upvote or should I have left the answer alone? Children's fantasy series with a symbol consisting of a circle divided into quarters I read this book series as a kid back in the late 1970s, so it's an old series - and my memories are similarly old, and vague.   I know it's a fantasy series. I think there are 4 books in the series. Or perhaps 7.  There's a central boy character who has to get a collection of magical objects (every fantasy needs a quest!), each of which represents a different magical 'element': wood, stone, fire, etc. I definitely remember there was a wooden object. He lives in our normal world. The whole series takes place "here &amp; now". I have a feeling he's a "chosen" person of some kind... But, the strongest memory I have is of a symbol: a circle which is divided into quarters (a vertical bar and a horizontal bar).  I know it's vague, but does anyone recognise this?