Consider a sequence of $n$ Bernoulli trials with $P(\text{success})=p$. Let $X_i$ and $X_j$ be indicator variables of the number of "success" in $i$th and $j$th runs with $i&lt;j$. In other words $X_i=1$ if the $ith$ trial is success and $0$ otherwise.. Given the total number of success was $m$. I am asked to compute the correlation coefficient for $X_i$ and $X_j$.  To compute the co-variance I will need to find $E(X_i)$ and $E(X_j)$ first. By definition $E(X_i)=p$ =$E(X_j)$. But how should I find E($X^2_i$)?
Consider a sequence of $n$ Bernoulli trials with $P(\text{success})=p$. Let $X_i$ and $X_j$ be indicator variables of the number of "success" in $i$th and $j$th runs. Given the total number of success was $m$, $m&lt;n$. I am asked to compute the correlation coefficient for $X_i$ and $X_j$.  Now I know the formula for correlation coefficient, but to compute the co-variance I will need to find $E(X_i)$ and $E(X_j)$ first. By definition does $E(X_i)=p_i$ and $E(X_j)=p_j$? I feel this seems a bit too easy but I can't see whats wrong with it.
Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$ Why does boiling water in the microwave make a cup of tea go weird? When I boil water in the kettle, it makes a nice cup of tea.  Sometimes I need to use a microwave because a kettle isn't available.  I boil the water in the mug and it looks pretty normal, but when I drop in the teabag the water froths up and looks foamy.  I don't see what the chemical difference is here, so I assume it must be some physical difference.  I have noticed this with multiple types of tea and multiple microwaves, the results being consistent so it's not just a weird microwave or something like that.    What is the reaction here and how/why does it occur?  Here is a photo of the 'fizzy' looking tea just after dunking in the teabag. It was suggested that I should get myself a polarizer filter to get over my reflection problem. (  )  I found out that a Nikon 52mm Circular Polarizing Filter CPL   while I can get another filter for less than half the price — . (Please note that I know nothing about these filters and googled them just for the example.)  What is the difference between polarizer filters?  I need to remove the glare and nothing else — would either of the two do the magic for me? What filter do you recommend for me? Maximum total distance between points on a sphere What is the configuration (set of locations) of $n$ points on the surface of a sphere such that the sum of distances is maximum for $n=1,2,3,...$?  The sum of distances is measured by summing the lengths of every straight line (through the sphere) connecting every possible combination of $2$ points.  All the points are on a single sphere of radius $R$.  Here's a visualization:   Acknowledgements: Based on  Physics S.E. question. Image from . Why does a bigger sensor lead to a shallower depth of field? It has been mentioned at    that a larger sensor results in images with a shallower depth of field. Example image:          I understand how sensor size would relate to, for example, field of view, but the relationship with depth of field does not seem straightforward.  It actually seems contradictory - I have more wells on my sensor, and I am able to focus on less number of points.  What is the reason for this effect? I have some thoughts on this. First, I want to say I am no expert on cryptography, I just know some stuff, and I took a cryptography class in University. I am very interested in this topic.  I understand that many security experts out there might get angry just by the discussion of these things, but I hope you can spare me :)  My Questions:  Would it be possible to break an RSA key, in for example 1 week of time, if the cracker have already spent X number of years building an index of primes by performing every permutation of existing prime keys up to 2^2048 ?  I understand this would take an immense amount of time to do, but it would only be done once.  Given the public key, you would be able to look up what the two primes are, and hence retrieve the private key instantly.  When I consider who might break such a code, I am primarily thinking about NSA and other Governments that might have spent years in research and have the resources and interest in doing so.  Breaking an 256 AES would then be easier, as AES keys are often generated using the RSA private key.  Would it be possible to build such an index ? How long would that take in permutations, and perhaps with the fastest computer in the world ?  I hope we can have a nice chat about this. We are just discussion theory here. I hope you can offer me some insight on this topic.  Thanks! Determine $\text{Aut}(S)$ where $S = \mathbb{Q}[x]/(x^2)$ Here's the problem: if $S = \mathbb{Q}[x]/(x^2)$, compute the group  of automorphisms $\mbox{Aut}(S)$.   First off, I have the solution and I know it's isomorphic to the multiplicative group $\mathbb{Q}\backslash\{0\}$, so I'm trying to understand it. It uses what is called the substitution principle and the fact that the restriction of any endomorphism $\phi: \mathbb{Q}[x]\to \mathbb{Q}[x]$ to $\mathbb{Q}$ is the identity. It says that these two facts implies that the endomorphisms of $\mathbb{Q}[x]$ are in 1-1 correspondence with $h \in \mathbb{Q}[x]$, where the endomorphism corresponding to $h$ is $\phi_h : \mathbb{Q}[x] \to \mathbb{Q}[x]$ given by $\phi_h(f) = f(h)$ for $f \in \mathbb{Q}[x]$. I guess I just don't understand how it applied the substitution principle? It's probably obvious, but for some reason I am not getting it.  An explanation would be greatly appreciated! Or if you have a different solution, that'd be cool too. Thanks!! The "Very Low Quality" (VLQ) flag reason is really annoying me.   It is the analog to the mommy mommy flag (which my 1.5-year-old child raises quite frequently). It provides no real clue as to what the flagger really wants us to do with the post.   Does she want us to close the question?  Does she want us to edit it into shape?  Does she want us to delete the question?  Does she want us to ban the user?  Does she want us to jump around and have a party?    Should we get rid of this flag reason?   Is it just me? Should and can we replace it with something better?     EDIT I accepted Jeff's answer. The tougher wording has heavily decreased the amount of very low quality flags. We now see only about 25% of the amount we used to and they are usually valid. Additionally, we now only allow very low quality on posts with a score of 0 or lower. Capturing console output from a .NET application (C#) How do I invoke a console application from my .NET application and capture all the output generated in the console?   (Remember, I don't want to save the information first in a file and then relist as I would love to receive it as live.) What is meant by "object serialization"? Can you please explain it with some examples? I'd like to get a notification when a question that I've voted to close (but isn't closed yet) is edited Equivalence Relation with dividing x and y integers Define $x\sim y$ means 5 divides $(x - y)$ for $x$ and $y$ integers. Show that is an equivalence relation.  Equivalence relation means it satisfies reflexity, symmetry, and transitivity.  reflexive: $x\sim x$ means 5 divides $x$  symmetry: $x \sim y \rightarrow y\sim x$ means 5 divides $x - y$ and 5 divides $y - x$: $$5/(x - y) = 5/(y - x)$$ so symmetry is satisfied.  I am not sure if I am right here and I am lost on how to prove it is transitive any suggestions would be greatly appreciated What's a good phrase for "refining a process which is hopelessly broken"? I'm looking for a turn of phrase to describe a situation where the powers that be wish to continue making small improvements to a process which, due to deep-rooted flaws, will never be close to accurate.  I'm working on a process for estimating the states of business objects, and there's interest in fixing some edge cases which have been noticed.  This could be worthwhile, but because of missing data we're already making heroic guesses.  So if implemented, we might get from 80.0 to 80.1% accuracy.  If we were at 99.5%, I'd be happy to get it to 99.6%, but in the current situation it doesn't seem productive.  We have some CS people on E.SE; what would you call it when someone wants to optimize a two-second process to load data, though once loaded it will be chewed on by an inefficient algorithm for hours?  Something like "bailing the ocean" or "fixing a leaky faucet in a burning building" is close to what I have in mind.  There's also "rearranging deck chairs on the Titanic," but that carries an implication that the changes are wholly useless, and that events are coming to a head. How to derive variance-covariance matrix of coefficients in linear regression I am reading a book on linear regression and have some trouble understanding the variance-covariance matrix of $\mathbf{b}$:    The diagonal items are easy enough, but the off-diagonal ones are a bit more difficult, what puzzles me is that  $$ \sigma(b_0, b_1) = E(b_0 b_1) - E(b_0)E(b_1) = E(b_0 b_1) - \beta_0 \beta_1 $$  but there is no trace of $\beta_0$ and $\beta_1$ here. How to change the list of file types which Safari will open immediately after download? I would like to change Safari Safe List of type files which open immediately after download. Any ideas how? It seems that OTP encryption is unbreakable. I have read some articles (Wikipedia, Blogpost, etc.), but I could not find any information about how they are used.  If I wanted to create a messenger service, how would I distribute the OTP keys? Both the sender and the receiver need this key before the message is sent, right?   And how do I make sure no one else gets these keys? When testing the following code:  from osgeo import gdal import json   def read_file(filename):     f = gdal.VSIFOpenL(filename, "rb")     if f is None:         return None     content = gdal.VSIFReadL(1, 10000, f).decode('UTF-8')     gdal.VSIFCloseL(f)     return content   gdal.VectorTranslate('out.json', """{   "type": "FeatureCollection",   "features": [       { "type": "Feature", "geometry": { "type": "Polygon", "coordinates": "coordinates":[[[-188.56933593749997,61.58549218152362],[-173.9794921875,61.58549218152362],[-173.9794921875,66.00015035652659],[-188.56933593749997,66.00015035652659],[-188.56933593749997,61.58549218152362]]] } },   ] }""", format = 'GeoJSON', layerCreationOptions = ['RFC7946=YES', 'WRITE_BBOX=YES'])  got = read_file('/vsimem/out.json') gdal.Unlink('/vsimem/out.json')  print json.loads(got)   from this , the file out.json is empty. The folder is with all permissions for writing allowed. When testing the method gdal.OpenEx, is possible to load an object, but still the method gdal.VectorTranslate returns an empty geojson when trying to pass this new loaded object as a srcDS object. Any hint on solving this issue? This was a question from my textbook. This is how I approached it.   For the first part, you can do c(10,5) so you choose 5 of the 10 to be on a team.   For the second part, you have 10 ways to choose the "best player" and you multiply it by c(10,4) to pick the other 4 players. However, I am not sure how you can exclude the "worst player". Although this sounds subjective, I believe the question is saying that the "best" and "worst" players are distinct, whereas the other players aren't. Alpha and Omega limit sets (dynamical systems) What are all the possible $α$- and $ω$-limit sets of points for the linear system of equations $x'=Ax$ given by the following matrices $A$:  I guess I don't really know how to get started, Could I have some guidance? So I was told to calculate the eigenvalues (fine) and eigenvectors (I kind of forgot how to do that...)  I know that the $ω$-limit sets of points are the set of points that the system of equation approach as time goes to infinity, and the $α$-limit sets of points are the points approached as t goes to negative infinity.   $    A=   \left[ {\begin{array}{cc}    -4 &amp; -2 \\    3 &amp; 11 \\   \end{array} } \right] $   $λ_1=-10$ $λ_2=-5$  Since both eigenvalue are negative, this is a stable node, and the $ω$-limit set=0.  for $λ_1=-10$:  eigenvector $    =   \left[ {\begin{array}{cc}   1 \\    3 \\   \end{array} } \right] $  for $λ_2=-5$:  eigenvector $    =   \left[ {\begin{array}{cc}   2 \\    1 \\   \end{array} } \right] $   $    A=   \left[ {\begin{array}{cc}    1 &amp; 1 \\    3 &amp; 1 \\   \end{array} } \right] $   $λ_1=4$ $λ_2=-2$  Since one eigenvalue is positive and one is negative, it is not stable, so I can't use the same reasoning that I used above to find the $ω$-limit set... so I am not sure how to continue  for $λ_1=4$:  eigenvector $    =   \left[ {\begin{array}{cc}   1 \\    1 \\   \end{array} } \right] $  for $λ_2=-2$:  eigenvector $    =   \left[ {\begin{array}{cc}   -1 \\    1 \\   \end{array} } \right] $   $    A=   \left[ {\begin{array}{cc}    4 &amp; 5 \\    -5 &amp; -2 \\   \end{array} } \right] $   $λ_1=1+4i$ $λ_2=1-4i$  for $λ_1=1+4i$:  eigenvector $    =   \left[ {\begin{array}{cc}   -3/5-(4i)/5 \\    1 \\   \end{array} } \right] $  for $λ_1=1-4i$:  eigenvector $    =   \left[ {\begin{array}{cc}   -3/5+(4i)/5 \\    1 \\   \end{array} } \right]$ Keep processes alive after SSH logout Im trying to measure bandwidth, packet loss, latency and jitter between two computers on a wireless 802.11s mesh connection.   Im trying to make a (my first) bash script that can do the following:   SSH to another computer and start logging data from services (ptpd, tcpdump, iperf). Start the same processes on my own computer and start logging data. Once the iperf transmission on my own computer is finished, it should SSH into the remote computer and kill the the services.   The idea is for this these three steps run in a for-loop, that increases the package size transmitted by iperf for each iteration and logs the resulting data.  I know how to start a process on the remote pc by ssh host@adress process.  I am however clueless on how to start multiple processes through SSH and logout, since the processes are terminated once I logout. I have looked at the screen and nohup commands, but don't know how to use them in a script.  Thanks in advance!  EDIT:  Thanks for all the help so far.  I got nohup to work now, it was pretty simple as people pointed out.  So now I got: ssh host@adress nohup ./script.sh &gt; /dev/null 2&gt;&amp;1 &amp; in the script on my main pc. This starts a script on the remote pc just fine, and then logout of SSH. The next problem is how to execute the script on the remote pc as sudo, since this is needed by tcpdump.  Trying ssh host@adress **sudo** nohup ./script.sh &gt; /dev/null 2&gt;&amp;1 &amp; doesnt work.  Any ideas?  EDIT 2:  If anybody is stuck with the same problem, I found a guide how to solve this: