100 participants have a fair coin each, on a given round, the not already discarded participants flip their coins, those who flip a tail are discarded from the game, the remaining ones continue to play until nobody is left (everyone has been discarded).  What would be the average number of trials (where each trial consists of a tossing and removing the tails) one would expect from doing this experiment?  Does conditional expectation work for something like this?   I know that each individual coin follows a Geometric distribution, but I am trying to figure out the sum of them to determine the average number of trials for a game like this. My Logic/Thought Process: I started out trying to think of the probability that a particular coin makes it to round $r$ which is $\frac{1}{2^m}$. I then realized that each coin outcome can be modeled by a Geometric random variables with $p = 0.5$. I am just now unsure how to take the leap from this single case to a case with 100 coins. I presume it has to do with summing the geometric random variables, but I am not sure.
Given $n$ independent geometric random variables $X_n$, each with probability parameter $p$ (and thus expectation $E\left(X_n\right) = \frac{1}{p}$), what is $$E_n = E\left(\max_{i \in 1 .. n}X_n\right)$$    If we instead look at a continuous-time analogue, e.g. exponential random variables $Y_n$ with rate parameter $\lambda$, this is simple: $$E\left(\max_{i \in 1 .. n}Y_n\right) = \sum_{i=1}^n\frac{1}{i\lambda}$$  (I think this is right... that's the time for the first plus the time for the second plus ... plus the time for the last.)  However, I can't find something similarly nice for the discrete-time case.    What I have done is to construct a Markov chain modelling the number of the $X_n$ that haven't yet "hit". (i.e. at each time interval, perform a binomial trial on the number of $X_n$ remaining to see which "hit", and then move to the number that didn't "hit".) This gives $$E_n = 1 + \sum_{i=0}^n \left(\begin{matrix}n\\i\end{matrix}\right)p^{n-i}(1-p)^iE_i$$ which gives the correct answer, but is a nightmare of recursion to calculate. I'm hoping for something in a shorter form.
Imagine that you want to develop a non-trivial end-user desktop (not web) application in Python. What is the best way to structure the project's folder hierarchy?  Desirable features are ease of maintenance, IDE-friendliness, suitability for source control branching/merging, and easy generation of install packages.  In particular:   Where do you put the source? Where do you put application startup scripts? Where do you put the IDE project cruft? Where do you put the unit/acceptance tests? Where do you put non-Python data such as config files? Where do you put non-Python sources such as C++ for pyd/so binary extension modules? How to divide two 64-bit numbers in Linux Kernel? Some code that rounds up the division to demonstrate (C-syntax):  #define SINT64 long long int #define SINT32 long int  SINT64 divRound(SINT64 dividend, SINT64 divisor) {   SINT32 quotient1 = dividend / divisor;    SINT32 modResult = dividend % divisor;   SINT32 multResult = modResult * 2;   SINT32 quotient2 = multResult / divisor;    SINT64 result = quotient1 + quotient2;    return ( result ); }   Now, if this were User-space we probably wouldn't even notice that our compiler is generating code for those operators (e.g. divdi3() for division). Chances are we link with libgcc without even knowing it. The problem is that Kernel-space is different (e.g. no libgcc). What to do?  Crawl Google for a while, notice that pretty much everyone addresses the unsigned variant:  #define UINT64 long long int #define UINT32 long int  UINT64 divRound(UINT64 dividend, UINT64 divisor) {   UINT32 quotient1 = dividend / divisor;    UINT32 modResult = dividend % divisor;   UINT32 multResult = modResult * 2;   UINT32 quotient2 = multResult / divisor;    UINT64 result = quotient1 + quotient2;    return ( result ); }   I know how to fix this one: Override udivdi3() and umoddi3() with do_div() from asm/div64.h. Done right? Wrong. Signed is not the same as unsigned, sdivdi3() does not simply call udivdi3(), they are separate functions for a reason.  Have you solved this problem? Do you know of a library that will help me do this? I'm really stuck so whatever you might see here that I just don't right now would be really helpful.  Thanks, Chad Phone keeps rebooting on Volume Up + Power (trying to install ROM) I tried to install Android's ROM and got ERROR 7, then I wiped out cache, data, internal storage, where was twrp.apk installed. After that I cant get into boot menu (with Install, Wipe... buttons). Then I copied twrp to the phone: sudo fastboot flash recovery &lt;twrp.img&gt; Tried versions 3.2.1-0 and 3.4.0-0, because I heard for certain phone (I have Xiaomi Mi4) it's necessary to use proper version, though I don't know which. What's going on now: I can successfully can get to the nice splash screen using Volume Down + Power, then on my PC I give sudo fastboot reboot command and press Volume Up + Power and here is the problem: It has no effect, it just keeps rebooting (I see only Xiaomi's logo). Problem  For what natural numbers is $n^3 &lt; 2^n$?  Attempt @ Solution   For $n=1$, $1 &lt; 2$ Suppose $n^3 &lt; 2^n$ for some $n = k \ge 1$ It looks like the inequality is true for $n = 0$, $n = 1$ and $n\ge10$ But, how can I prove this through induction? Moving from HTTP to HTTPS in Google Search Console I'm moving from HTTP to HTTPS for my entire site. The site is currently added to Google Search Console (formerly Google Webmaster Tools) as www.example.com and all the pages are indexed without using SSL.  How do I go about moving to the new HTTPS URLs on Google Search Console?   Do I just submit an updated sitemap which has the HTTPS URLs or do I add a new site as https://www.example.com and submit the sitemap with HTTPS URLs? All current URLs are set to redirect to their HTTPS counterparts. Do I add multiple properties to Google Search Console? How to use early stopping properly for training deep neural network? I have a deep neural network model and I need to train it on my dataset which consists of about 100,000 examples, my validation data contains about 1000 examples. Because it takes time to train each example (around 0.5s for each example) and in order to avoid overfitting, I would like to apply early stopping to prevent unnecessary computation. But I am not sure how to properly train my neural network with early stopping, several things I do not quite understand now:   What would be a good validation frequency? Should I check my model on the validation data at the end of each epoch? (My batch size is 1) Is it the case that the first few epochs might yield worse result before it starts converging to better value? In that case, should we train our network for several epochs before checking for early stopping? How to handle the case when the validation loss might go up and down? In that case, early stopping might prevent my model from learning further, right?   Thank you in advance. I find in many places mentions of pre-amplifiers being used to boost audio signals prior to feeding to a power amplifier.  Wikipedia's article on  gives some hints at their usage, but it doesn't really explain why it would be necessary to have two amplifiers in series rather than a single one with greater gain.  If a pre-amp is just a power amp with a smaller gain, what's the point, and why the distinction between the two kinds? Is the distinction in function, or simply purpose? How can I automatically shutdown the system after a specific time? amsalpha-like bibtex style with manual labels Posts linked in deleted answers still show up under "Linked" E.g. see . How to prevent ENTER keypress to submit a web form? Using SQLite3 with Python 2.5, I'm trying to iterate through a list and pull the weight of an item from the database based on the item's name.  I tried using the "?" parameter substitution suggested to prevent SQL injections but it doesn't work. For example, when I use:  for item in self.inventory_names:     self.cursor.execute("SELECT weight FROM Equipment WHERE name = ?", item)     self.cursor.close()   I get the error:     sqlite3.ProgrammingError: Incorrect number of bindings supplied. The current statement uses 1, and there are 8 supplied.   I believe this is somehow caused by the initial creation of the database; the module I made that actually creates the DB does have 8 bindings.  cursor.execute("""CREATE TABLE Equipment      (id INTEGER PRIMARY KEY,      name TEXT,     price INTEGER,      weight REAL,      info TEXT,      ammo_cap INTEGER,      availability_west TEXT,     availability_east TEXT)""")   However, when I use the less-secure "%s" substitution for each item name, it works just fine. Like so:  for item in self.inventory_names:     self.cursor.execute("SELECT weight FROM Equipment WHERE name = '%s'" % item)     self.cursor.close()   I can't figure out why it thinks I have 8 bindins when I'm only calling one. How can I fix it? Installing Ubuntu with Fake RAID 1 How do berry mutations work? In Ubuntu 18.04.1, I would like to use my left Super-key only to simply open the Show Applications menu (aka Dash) like a left click on the dock-icon (with the 9 dots) does and preferably close it again without showing me any open programs.   I can open it with Super + A and even close it again with Super R, but Super L appears to do nothing at all.   In Xfce, something similar was possible by adding a custom keyboard shortcut for Super L and setting the command to xfce4-popup-whiskermenu, which would open and close a menu with favourite applications. Unfortunately, I don't know the command that opens and closes the Show Applications menu.   Is there any way to add such a shortcut in the new Gnome Shell? What is the location of lock screen images on Windows 10? Where, or what format, are the images used by Windows 10 when logged out or in lock-screen?  I searched the entire drive for *.jpg, *.bmp, *.png, and *.tif but the one I was looking for did not appear.  At first glance, , although not exactly a duplicate, appeared to have the path I wanted.  But when I went there, the pre-login image was there, but the lock-screen image was not.  (And contrary to the claim in that answer, the files there did have extensions.)  UPDATE: I just did a lockscreen, and got an image that IS in that directory, but earlier today, it was consistently an image of an interesting footbridge that is not in that directory. What is the difference between SAN, NAS and DAS? What is the difference between SAN, NAS and DAS? Why is it so difficult to find the twitter account for a Stack Exchange site? I am not able to find a link to the twitter account of Academia SE (), for example, in the Academia SE main webpage.   Why are twitter links not included on the main site page of each Q&amp;A site in SE? If you have a single atom in a box, what state of matter would it be in? If the walls of the box were made of a completely inert material, what state of matter would the atom be in? Or would it not have one? I realized this is a duplicate of several other questions. Reference - What does this error mean in PHP?