In my statistics class, it was proven that the sum of independent and identically distributed random variables distributed according to an exponential distribution follows a gamma distribution using moment-generating functions. However, one thing that confuses me is how the formula for the Gamma distribution seems to come out of nowhere; i.e. the proof confirms that it is correct, but it does not seem to derive it from the exponential distribution definition.
Let $X_1$ and $X_2$ be independent and identically distributed exponential random variables with rate $\lambda$. Let $S_2 = X_1 + X_2$.  Q: Show that $S_2$ has PDF $f_{S_2}(x) = \lambda^2 x \text{e}^{-\lambda x},\, x\ge 0$.  Note that if events occurred according to a Poisson Process (PP) with rate $\lambda$, $S_2$ would represent the time of the 2nd event.   Alternate approaches are appreciated. The approaches provided are commonly used when learning queueing theory &amp; stochastic processes.     Recall the Exponential distribution is a special case of the Gamma distribution (with shape parameter $1$). I've learned there is a more general version of this  that can be applied.
How to set BibTeX bibliography title? Currently my bibliography says "Literatur" (german). But I want something completely different there. How can I achieve that? How to install Chrome browser properly via command line? I am trying to install chrome browser via command line.  I tried this:  $ apt-cache search chrome browser   The results show that the proper term is "chrome-browser," so I tried that:  $ sudo apt-get install chrome-browser   And then "Y" for the Y/n question.  But the installation threw errors.  Does someone see anything wrong with the commands I issued? Why didn't Isildur's Heir come forward sooner? regex error - nothing to repeat What does "yield break;" do in C#? I have seen this syntax in MSDN: , but I don't know what it does.  Does anyone know? How to integrate OpenCV Manager in Android App Error -504 on appliation installation Yes. I've read all other questions regarding this problem. Like this one:  It's the same problem as this one but with more background information and an other app:   The details: My device is an Ulefone Be Touch 2 running Android 5.0.  My connection is perfect. To be sure I tried different networks, among them are 3G, 4G and wifi on ADSL, VDSL, cable and fiber. I've also used Tor and 3 different VPN providers in 3 countries on 2 continents. Let's say I ruled the network out.   A single app does not install: OpenKeychain.  Currently I have 126 apps installed. It does not install via desktop browser > play store > install button.    I have enough free space (40GB). The app is 7MB.   Re-adding Google account and clearing caches did not work.    Even a complete device wipe and re-install didn't work.     Downloaded the apk and tried manual installation, same error.   I also own a Nexus 7 (2013 wifi) and the app installs fine on that device. Do MOSFETs have a diode built into them? I have noticed that a MOSFET symbol has a little diode in it (or at least what looks like a diode). Does this mean I do not have to worry about using a diode in a circuit that runs a motor using one? I would have used a diode in order to prevent reverse voltage from turning the motor by hand. Often while playing Pathfinder, my group will run into problems where rules for 3-dimensional play simply don't exist, so far as we can find. Arguments will come up over things like carrying-capacity, movement speed, area-effects, and a variety of other things.  Simply put, are there rules for 3D play for Pathfinder, and if so where can I find them? I'm using BibLaTeX to handle my bibliography and I want a reference to be placed in a footnote on every page where I cite the same work and have the same number. Also I want to be able to cite the same work on one page several times. The latter could be done by using  from the Joseph Wright's blog. However, it puts a reference in a footnote only once. As mentioned in the , Ubuntu 18.04 LTS will not have a 32 bit Desktop Image Installer. But in the  it has  32bit Desktop Images.  This is little off topic, but I'm confused. Morphing only a part of a full path in tikz This might be a tikz newbie question but I have some trouble solving this.  I want to draw and fill a rectangular path where path morphing should be applied only to the upper side of the shape, more specifically I want to have an upper part with random steps. The rest of the path (left, lower and right sides) should just be straight lines. The shape should be closed and filled with one color.   My problem is that I cannot figure out how to apply the morphing only to the upper side and I suppose I cannot break up the path into different subpaths since I want to fill the final shape. Is there a way to use the morphing only on a part of the full path? Another problem is that the cycle command does not really close the shape at the upper left corner.  I tried to find a solution in the tikz manual but there was nothing that really helped me.  Here is some example code:  \documentclass[tikz, border=10pt]{standalone} \usepackage{tikz}  \begin{document}  \usetikzlibrary{decorations.pathmorphing} \begin{tikzpicture}  \draw[decorate, decoration={random steps,segment length=3.0mm, amplitude=2.0mm}, thick, rounded corners=1ex, fill=brown!70] (0,0) -- (4,0) -- ++(0, -1) -- ++(-4,0) -- cycle;   \end{tikzpicture} \end{document} I need a formula for conditional formatting to make an Excel 2010 spreadsheet alternate highlighting when a single column changes.  This question, "", is exactly what I want to do, but the suggested solution didn't work for me--it just highlights the first row of each matching value.  I know how to use conditional formatting, but for the life of me I can't figure out or find any pointers on the net so far to make this work.  Ideas?  Here is a link to a picture of how I want my spreadsheet to look when I'm done.  Basically I want every other Disp Number value row to be highlighted.  Seems like it would be a common thing to want to do, and I've seen this asked for in various places, but people struggle with making it work. Picking coefficient in Inverse Distance Weighted method of QGIS? I'm using QGIS 1.8.0 to interpolate weather data. This is, I have a weather station located in say, A and C. With the Inverted Distance Weighting or Triangular Interpolation methods, I can know/interpolate information in B. When using the IDW method, I have to choose a coefficient.   Does anyone know in which range the coefficients should be?   Say from 1 to 10 or .1 to .99 or 100 to 1000?   Any ideas? Let $(e_n)$ (where $ e_n $ has a 1 in the $n$-th place and zeros otherwise) be unit  standard vectors of $\ell_\infty$.   Why is $(e_n)$ not a basis for $\ell_\infty$?  Thanks. A potential customer wants to evaluate our storage system. They run Windows 2008R2 x64 with 4KB NTFS on a virtual test machine they've sent us. They did not seem to know this at hand, so I think it's reasonable to assume that environment was not tweaked. The tests are inserting, indexing, searching, deleting. Tool is not known to me.  Given that the windows NTFS block size is 4KB, and SQL write in 64KB chunks - is it safe to assume that a block size of 64k on the SAN is a good choice?  They run SQL Server 2008, perhaps standard. Determine when $\sum_1^\infty \frac{(2n)!x^n}{n(n!)^2}$ converges. By ratio test, when $|x|&lt;1/4$, the sum converges, $|x|&gt;1/4$ diverges. But I'm not sure about $|x|=1/4$. By Stirling's approximation $\frac{(2n)!(1/4)^n}{n(n!)^2}\sim\frac{4^n}{n{4^n}\sqrt{ \pi n}}\sim\frac{1}{\sqrt{\pi}n^{3/2}}$, so when $|x|=1/4$, the sum converges. But I haven't learned Stirling's approximation yet, so how to prove that when $|x|=1/4,$ the sum converges, without using it? I've been using the biblatex package for a while, and since updating my TeX installation (post-el capitan upgrade) I have been experiencing the following.   If the following is typed:  Such and such has been said \citep[111]{citekey}.   The following text is displayed:  Such and such has been said (Author, 2016, p. 111).   And if the following is typed:  \citet[111]{citekey} has said such and such.   The following text is displayed:  Author, (2016, p. 111) has said such and such.   The citep command is working as I want it to, but I do not want a comma placed after the author's name when using the citet command. I would like it to display as follows:  Author (2016, p. 111) has said such and such.    If I use the \citeyearpar command, it is fine, but I would rather not have to type the author's name in manually.   Here is the command placed in the preamble:  \usepackage[backend=bibtex, style=authoryear-icomp, url=true, isbn=false, natbib=true, sortcites=true, block=space, pagetracker=false, loccittracker=constrict]{biblatex}   Here is an example of an entry in my bib file (made using BibDesk):  @article{poldrack2006, Author = {Poldrack, Russell}, Journal = {Trends in Cognitive Sciences}, Number = {2}, Pages = {59--63}, Title = {{Can cognitive processes be inferred from neuroimaging data?}}, Volume = {10}, Year = {2006}}   I have also tried changing the entry to the following:  @article{poldrack2006, Author = {Russell Poldrack}, Journal = {Trends in Cognitive Sciences}, Number = {2}, Pages = {59--63}, Title = {{Can cognitive processes be inferred from neuroimaging data?}}, Volume = {10}, Year = {2006}}   The bibliography entry appears as follows, and is fine:  Poldrack, Russell (2006). “Can cognitive processes be inferred from neuroimaging data?” In: Trends in Cognitive Sciences 10.2, pp. 59–63.  Thank you for any help that can be offered. Suppose that $f$ is continous on $[0,1]$ , differentiable on $(0,1)$ , and $f(0)=0$ and  $f(1)=1$.For every integer $n$ show  that there must exist $n$ distinct points $\alpha_1,\alpha_2\cdots,\alpha_n$ in that interval so that    $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $   Is it theorem or i can show it with basic differentiation knowledge .Please give me hint . At what time will the upcoming Ubuntu release be made available?