I understand that we can calculate the probability density function (PDF) by computing the derivative of the cumulative distribution formula (CDF), since the CDF is the antiderivative of the PDF. I get the intuition for that (integrals denote the area under a curve, which is the accumulated probability under the curve of continuous functions). I'm just wondering how to derive the CDF from the PDF of the Gaussian distribution, which is $$ f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{- \frac{1}{2} \left(\frac{x - \mu}{\sigma} \right)^2} $$ I'm guessing we integrate this from negative infinity to positive infinity, but how do we do the whole process? I've seen the answer, and for some reason the answer seems to involve the &quot;error&quot; function, which is this: $$ \text{erf} (z) = \frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} dt $$ If someone could explain how we derive the CDF from the PDF of Gaussian distribution, AND how that calculated CDF is related to the error function, I would be so grateful!
If the Standard Normal PDF is $$f(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$$  and the CDF is  $$F(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-x^2/2}\mathrm{d}x\,,$$  how does this turn into an error function of $z$?
I'm told by smart people that $$0.999999999\ldots=1$$ and I believe them, but is there a proof that explains why this is? Limit $\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$ How can I find the following limit :$$\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\left(1-\frac{1}{10}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$$There are four options $(a)\frac1{3}(b)\frac1{9}(c)\frac1{81}(d)0$ Thanks! As the title says. How do I get from  $\frac{n}{\varphi(n)}$ to $\sum\limits_{d \mid n} \frac{\mu^2(d)}{\varphi(d)}$?  I know that $$\frac{n}{\varphi(n)}=\frac{\sum\limits_{d \mid n} \varphi(d)}{\sum\limits_{d \mid n} \mu(d)\frac{n}{d}},$$ and I suspect it is down this road I should go. But I get totally confused by the sums and I have no clue on how to divide them and "combine" them into one again. Any hints or advice would be much appreciated. I'd like to create a Yahoo! account. The signup page won't let me proceed without entering a cell phone number, and it won't recognize my Google Voice number as a valid cell phone number. Is there a workaround to be able to create a Yahoo! account without having a cell phone that can receive SMS messages? Integral of $\frac{\sqrt{x^2+1}}{x}$ So I have to do the integration of $\frac{\sqrt{x^2+1}}{x}$. Give me a hint. What should I replace? Should I do it with integration by parts? Since I upgraded to Windows 10, my system has been consuming RAM excessively    I've been reading a bit and determined it's likely a driver leaking memory. So I got myself the Windows Driver Kit and tracked memory usage with poolmon:    However, I don't really know how to proceed from here. Is the item tagged "smNp" the culprit in this issue? How do I go from there to actually identifying the driver?  I tried some stuff like "C:\Windows\System32\drivers>findstr /s smnp ." but it returned no results. I also took a look at the pooltag.txt file and this is the description I found for it:      So yeah, any help would be appreciated. Thanks in advance. When is the past perfect exactly needed? How much money do I need to show as proof of support when applying for a UK Standard Visitor visa? I am an Indian citizen studying in Italy on an Italian visa that expires 19 November, 2015. I have permesso di soggiorno. I want to visit the United Kingdom for my vacations for a week during March. I know the procedures already, however I am a little bit confused about the amount that I must show as a proof of support with my bank statements. I will stay for a week with my friends who are also students in Italy and will be in London for 5 days. What would be the amount in Euros that I need to show for my visit to UK? Why is 'ls' suddenly wrapping items with spaces in single quotes? I just noticed that on one of my machines (running Debian Sid) whenever I type ls any file name with spaces has single quotes surrounding it. I immediately checked my aliases, only to find them intact. wyatt@debian630:~/testdir$ ls 'test 1.txt'  test1.txt wyatt@debian630:~/testdir$ alias alias ls='ls --color=auto' alias wget='wget --content-disposition' wyatt@debian630:~/testdir$   Another test, with files containing single quotes in their names (also answering a request by jimmij): wyatt@debian630:~/testdir$ ls 'test 1.txt'  test1.txt  'thishasasinglequotehere'\''.txt' wyatt@debian630:~/testdir$ touch &quot;'test 1.txt'&quot; wyatt@debian630:~/testdir$ ls ''\''test 1.txt'\'''  test1.txt 'test 1.txt'          'thishasasinglequotehere'\''.txt'   update with new coreutils-8.26 output (which is admittedly much less confusing, but still irritating to have by default). Thanks to Pádraig Brady for this printout: $ ls &quot;'test 1.txt'&quot;   test1.txt 'test 1.txt'    &quot;thishasasinglequotehere'.txt&quot;  $ ls -N 'test 1.txt'  test1.txt test 1.txt    thishasasinglequotehere'.txt  Why is this happening? How do I stop it properly? To be clear, I myself set ls to automatically color output. It just never put quotes around things before. I'm running bash and coreutils 8.25. Any way to fix this without a recompile? EDIT: Appears the coreutils developers  to break with the convention and make this the global default.  UPDATE - October 2017 - Debian Sid has re-enabled the shell escape quoting by default.  And at the bottom of the reply chain to the previous bug report, &quot;the change was intentional and will remain.&quot;  I thought this had already been settled, but apparently it was just reverted so that the &quot;stable&quot; Debian branch could keep its &quot;feature freeze&quot; while getting the other fixes, etc. from the newer version. So that's a shame (in my opinion). UPDATE: April 2019: Just found a  that was caused by this change to ls. When you're confusing developers and generating false bug reports, I think it might be time to re-evaluate your changes. Update: Android toybox ls is now doing something similar to this but with backslashes instead of quotes. Using the -q option makes spaces render as 'question mark characters' (I have not checked what they are, since they're obviously not spaces), so the only fix I have found so far without rooting the device in question is to add this to a script and source it when launching a shell. This function makes ls use columns if in a terminal and otherwise print one-per-line, while tricking ls into printing spaces verbatim because it's running through a pipe. ls() {     # only way I can stop ls from escaping with backslashes     if [ -t 1 ]; then         /system/bin/ls -C $@ |cat     else         /system/bin/ls $@ |cat     fi } ls: cannot access .gvfs: Permission denied I used to work as a consultant and have been spoiled with nice looking slides. For instance, being able to create slides that have 2, 3, or 4 graphs (perfectly positioned and with nice headers!) at the click of a mouse button via macros.  Similarly, there are many other kinds of word slides that had multiple columns, etc. that would be nice to recreate as well as objects like "main point" boxes at the bottom that summarize a main point and ways to add a transparent highlight to any part of the screen.  Is this kind of functionality far beyond anything that LaTeX presentations can do or is there an archive somewhere of code for making more intricate presentations? How do I select an object in the 3D viewport? Integration of $\frac {1}{1+\sin x}$ For $\int \frac {1}{1+\sin x} \mathrm{d}x$, my calculus textbook told me to multiply the integrand by $\frac {1- \sin x}{1- \sin x}$ to obtain: $$ \int \frac{1-\sin x}{1-\sin^2 x} \mathrm{d}x= \int \frac{1-\sin x}{\cos^2 x} \mathrm{d}x$$  From here, it is a matter of breaking up the integral to two very trivial integrals, $\int \sec^2x \mathrm{d}x$ and $\int \sec x \tan x \mathrm{d}x$, which are easily found via commonly known rules.  My question is whether there exists a way of finding $\int \frac {1}{1+\sin x} \mathrm{d}x$ without having to perform the multiplication by $1- \sin x$ in the numerator and denominator. As in the title, I must prove that: If a complex function $f$ is Entire, and there exist $k,R &gt; 0$, $n \in N$ such that $|f(z)| &gt; k|z|^n$ for all $|z| &gt; R$: Then f is a polynomial.  We've already had a similar question with the inequality reversed, and found that it had to be a polynomial with degree less than n. I was able to show that if it was a polynomial, then it had to have degree less than n, but not that it had to be a polynomial in the first place.  Once again, I'm face with the same problem. I assume this time, the degree of the polynomial will have degree greater than or equal to n, but how do I show that such $f$ must be a polynomial?  It's not like I can go through every other possible function there is, can I? I can show it doesn't work for exponentials, for example, but I just can't think of how to show that this ONLY works for a polynomial... How can I animate an object fading out? Delete an old Facebook account Two years ago somebody made a fanpage for our company, but that person doesn't work here anymore. We don't have any login or password. We like to see that account be deleted, please. Alignment issue for the last modified information in the "Top Questions" page In the  (homepage), the asked/modified X mins ago text was left-aligned next to the tags. Previously the text was right-aligned. Seems some alignment issue introduced on that page recently. Screenshot for reference:      Configuration Version     Operating System Windows 10 Pro   Mozilla Firefox 89.0 (64-bit)   Google Chrome 91.0.4472.101 (Official Build) (64-bit)     Update: I see the same alignment issue in all other Stack Exchange site's &quot;Top Questions&quot; page too. Real analysis - proof approach help If $AB = I$ then $BA = I$ How can I automatically shutdown the system after a specific time?