Suppose a gambler starts with one dollar and plays a game in which he or she wins one dollar with probability $p$ and loses one dollar with probability $1-p$. Let $f_n$ be the probability that he or she first becomes broke at time $n$ for $n = 0, 1, 2, \ldots$ Find a generating function for these probabilities?  Thanks :)
Suppose a gambler starts with one dollar and plays a game in which he or she wins one dollar with probability $p$ and loses one dollar with probability $1 - p$. Let $f_{n}$ be the probability that he or she first becomes broke at time $n$ for $n = 0, 1, 2\ldots$. Find a generating function for these probabilities.  Is it binomial distribution $?$ and then use the $\operatorname{mx}\left(s\right) = rx\left(\mathrm{e}^{s}\right)$ of the binomial distribution $?$. Help$\ldots!$. I am correcting my quiz.
Is there a way to get administrative access from Windows Subsystem Linux files to Windows files correctly? "Dear Sir or Madam" versus "To whom it may concern" When is it appropriate to use the terms Dear Sir or Madam and To whom it may concern?  The rules I was taught state that Dear Sir or Madam should be used when you're writing a letter to a person about something that person has direct involvement in (e.g. returning a defective product to a customer service department).  By the same rules, To whom it may concern would be used for situations in which the recipient is a third party to the topic of the letter (e.g. regarding a letter of reference or recommendation).  Was I taught correctly?  Is the rule actually something else?  Or is this one of those situations where many people have an opinion but there is no real rule? I accidentally opened too many menus (or windows) and I want to reset them to default.  It is really annoying, and effecting my animation.  Thanks Proving that $\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1)\times (n-k+1)}$ converges Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$ Unable to login to account in ubuntu 18.04 vmware workstation 15, after update I installed the latest updates from the software update center today (I update when ubuntu displays a notification), restarted the machine and now it's unable to boot properly. The best screen I can get is displayed below. If I resize the screen then it goes to black. If I restart the machine then it displays just purple.    Note that the screen is unresponsive, I can't click anything on it.  Other than SMBus Host Controller not enabled! I don't see any other warnings. Note that I don't know if this warning existed before the update.  I'm running ubuntu 18.04 in vwmare workstation 15 player (latest update). Any ideas on how to begin debugging this?    Update: Trying to boot Ubuntu, with Linux 4.15.0-50-generic instead of Ubuntu, with Linux 4.15.0-51-generic didn't help.  Update: Blacklisting i2c-piix4 according to  answer didn't remove the warning or fix the issue.  Update: Blacklisting i2c-piix4 according to  answer removed the warning but didn't fix the issue.  Update: The answer from Uzair helped me log into the account (run dpkg and update grub from recovery mode).  Answer:  answer helped me fully to resolve the issue. I don't understand the fix. If anyone does, put it as the answer and I'll mark it as such.  Regarding duplicity: The answer from another question did solve my issue but I don't think my question is a duplicate of that one. I began with a fresh install of 18.04 a few months ago, did several updates without an issue and only today experienced issues which are different than those described in the other question. I think we experienced different bugs with wayland and we both solved it by disabling it which is a destructive solution applicable to many questions. Is there a way to send a file using POST from a Python script? I saw this several times in different movies, but I don't understand what is this supposed to mean. The main idea is when the scene is about to change, for one second or so we still see the picture from the old scene, but the sound is already coming from from the upcoming scene. Why do they do this? What's the purpose of this technique? How should one maintain and update a MiKTeX installation? Every once in a while after updating MiKTeX, the installation gets into an inconsistent state. This might be a problem with the packages (an example is described in ), but sometimes packages are split or renamed. The cases I remember are when  and when miktex-etex-base-2,9 was renamed to miktex-etex-base-2.9 (if I recall correctly; I am sure it was either that or something very similar). In each case the MiKTeX Update process suggested to remove the "old" packages but it didn't give the user any hints about him having to manually add the respective replacement packages back in the package manager to get one's MiKTeX installation back into a usable state.  What are best practices for keeping MiKTeX up to date? Note that I'm not asking about installing new packages or fonts in general; there are other questions covering this area.   In which order should one execute the Update process and the Synchronization of the repository? , it is recommended to check the MiKTeX homepage before updating. What else should one do? When and in which order (and why) should one execute the "Refresh FNDB" and "Update Formats" processes within the MiKTeX Options menu? What else can happen? Other advice? Careers: What does "allows remote" mean? When a posting is marked "allows remote", does it mean work in the office some days, work from home some days?  Or does it mean you could actually live out of town and maybe fly in as needed?  The reason I ask is I'm really interested in public facing software rather than internal/enterprise stuff, and most of that seems to happen in a few very expensive cities.  Given it's so hard to hire in places like SV/NYC, it seems like it could be a good scenario for companies if a developer had a really solid list of accomplishments, but I don't know how many companies think this way. Is it possible to write any bounded continuous function as a uniform limit of smooth functions Is $C^\infty(\mathbb{R})\subset C_b(\mathbb{R})$ dense? I.e. is any continuous bounded function $f:\mathbb{R}\to\mathbb{R}$ the uniform limit of smooth functions?   On any bounded interval this is true since by Stone-Weierstrass polynomials are smooth and dense in the continuous functions. I guess the result is still true for $\mathbb{R}$ but I don't know how to prove it. Since I don't have any integrability conditions on $f$ the technique of convoluting with mollifiers is probably not applicable. What is a word that could define someone who likes to cause conflict? Approximating $\pi$ with least digits Do you a digit efficient way to approximate $\pi$? I mean representing many digits of $\pi$ using only a few numeric digits and some sort of equation. Maybe mathematical operations also count as penalty.  For example the well known $\frac{355}{113}$ is an approximation, but it gives only 7 correct digits by using 6 digits (113355) in the approximation itself. Can you make a better digit ratio?  EDIT: to clarify the "game" let's assume that each mathematical operation (+, sqrt, power, ...) also counts as one digit. Otherwise one could of course make artifical infinitely nested structures of operations only. And preferably let's stick to basic arithmetics and powers/roots only.  EDIT: true. logarithm of imaginary numbers provides an easy way. let's not use complex numbers since that's what I had in mind. something you can present to non-mathematicians :) Is the W statistic output by wilcox.test() in R the same as the U statistic? I've recently been reading about the Mann-Whitney U test. It turns out that to carry out this test in R you actually need to run a Wilcoxon test!  My question: is the W statistic of wilcox.test in R identical to the U statistic? Report sites that use Stack Exchange content without following attribution rules here NB The following is a community-generated list of websites that republish Stack Exchange content without attributing it properly. It is no longer being maintained, because the procedure for reporting such sites has changed; see the duplicate for more information. There are a number of license-violating clones of Stack Exchange sites popping up that use Stack Exchange's  data without following our Creative Commons attribution terms. Those terms are , and are also included as a .txt file in every data dump we produce.  Google now has  in search results. Also, you can :  The option to block a site appears when you click a search result and then navigate back to the search results page. Click the &quot;Block&quot; link next to that result to block all pages within the site's entire domain.   This post was created as an attempt to organize information originally posted . You may also be looking for . Given that   $x^x = y$; and  given some value for $y$   is there a way to expressly solve that equation for $x$? Free GIS workshops, tutorials, and applied learning material This Q&amp;A lists free as in $0 workshops in GIS and related fields that have their material available to view or download online. The list is not limited to FOSS GIS, as GISers usually use a combination of open and closed source tools to accomplish their tasks. Some of these tools are easier than others and some are better documented. This list will be a great benefit to the community especially for new comers to the field. General GIS   from   from   from   from   Open-source GIS   from   from   from   from   from   ESRI Products   from   from   Transportation These links include either theoretical or applied transportation knowledge in transportation planning          Disaster Management    Databases   from   Geo Statistics    from Possible Duplicates:             I heard a professor of mathematics explaining that the following conslusions are true.   $\textrm{False}\Rightarrow \textrm{True}$ $\textrm{False}\Rightarrow \textrm{False}$   Could someone please explain to me why both conclusions are true? I am especially interested in the first case. I'd like to hear two explanations if possible: One for mathematicians and one for non-mathematicians. :-) I am compiling the following MWE using XeLatex within TeXnicCenter and get a garbled equation number:  \documentclass{article} \usepackage{amsmath} \usepackage{polyglossia} \setmainlanguage{hebrew} \newfontfamily{\hebrewfont}{Arial}  \begin{document} \begin{equation}\label{eq:1} 2+2=4. \end{equation} \end{document}      What is going on here? Everytime I sudo apt-get update it always not complete and will say in the end  E: Read error - read (5: Input/output error) E: The package lists or status file could not be parsed or opened.   Same goes, when I try to install something and do sudo apt-get install &lt;some app&gt;  What should I do ? I admit, that my drive has bad sectors, could that be the reason? Is there anything else I can do?