Two independent random variables $X$ and $Y$ have standard normal distributions.  Find the density function of random variable $Z=X+Y$ using the convolution method.   shows this problem on page 4 but it seems to use some tricks that are, well, pretty tricky. In particular getting behind their third line is beyond me. Is there an alternate way to show this that is more straightforward perhaps even with more (unskipped) steps?
Let $X, Y$ be independent standard normal random variables.  We already know that $X+Y$ is normal with mean 0 and variance 2. However, I am trying to prove this result in a slightly different way than usual, i.e., using convolutions.  We can address the problem in the following way: the sum of the two variables has a density $h(x)$ given by the convolution product of the densities of the two variables, say $f$ and $g$, so that   $$ h(x) = \int_{-\infty}^{\infty} f(x-y)g(y)dy. $$  However,  $$ f(y) = g(y) = \frac{1}{\sqrt {2\pi}}\exp(-y^2/2). $$  Hence now the integral becomes:  $$ h(x) =\frac{1}{2\pi}\int_{-\infty}^{\infty}\exp(-(x-y)^2/2)\exp(-y^2/2)dy. $$  However, I am puzzled here. How should we go on from here? I cannot see how to transform this integral into the integral of a normal r.v of type $N(0,2)$. Thanks in advance for your help.
Who turned on the lightsaber? Spoilers for The Last Jedi:      Just before Snoke is slain,   he narrates Kylo Ren's thoughts/feelings (saying things like "he turns the lightsaber towards his enemy" and so on) we see the Skywalker lightsaber being manipulated through the Force to point towards and then slay Snoke. My question is, do we know for sure who was using to Force to use the Skywalker lightsaber in this scene?   It could have been   A) Kylo Ren was turning igniting the Skywalker lightsaber with the Force. Snoke assumed the thoughts/feelings he was sensing were directed at Rey but they weren't.   Or... B) Snoke was so preoccupied with Kylo's thoughts that he wasn't paying any attention to Rey's thoughts. Rey takes advantage of this and uses the Force to turn and ignite the Skywalker lightsaber to slay Snoke. This option is reinforced by the fact that  1) the Skywalker lightsaber then zooms into Rey's hand, not Kylo's.  2) after the ensuing scuffle, Kylo Ren reveals that he hasn't switched sides; he is still a believer in the First Order.  3) Kylo Ren even tells Hux that Rey slew Snoke. How to remove zsh alias? I can always find that in some articles the production of higgs decay written as &quot;$\rm H\to {ZZ}^*\to 4l$&quot;, &quot;$\rm H\to ZZ\to 4l$&quot;, &quot;$\rm H\to Z\gamma^*\to\ldots$” What dose it mean when some particles with a superscript *? Such as “$\rm Z^*, W^*, \gamma^*, \ldots$&quot;. What’s the difference between them and &quot;$\rm Z, W, \gamma$&quot;? Proof that any simple connected graph has at least 2 non-cut vertices. How to force restart an iPhone with broken home button Earlier today I was on my iPhone and it randomly froze so I turned it off. When I went to unlock it it wouldn't move. I couldn't unlock it or tap anything. So now I can only stay on the lock screen. I read on the Apple website that I should "force restart" it by pressing my home button and power button at the same time. The problem is my home button is stuck and doesn't work. What should I do? Short story set in the near future where all wars are settled on the moon using automated machines? Plot Details/Summary  Another one where I have only sketchy memories of the tale, I fear.   The setting for this one is the near future (from the writer's perspective, which I think was the 1970s).  Man's reached a point where bloodshed and loss of life are no longer necessary to settle conflicts.  Instead, all disputes are settled through an annual combat held on the surface of the moon, using the best combat machines each nation can come up with.  The protagonist, as I recall, is an engineer for the Americans.  He recollects past conflicts, and the circumstances for this years fracas.  Much of the combat is told from his observers' perspective.  The actual conflict is a rather clever affair, with a description of each team's designs and all the tricks and gee-whiz things they built into them.  The protagonist's team manages to prevail, when they managed to use an attack as a feint, with the real attack being a swarm of microscopic machines (they did not use the term nanites) that destroyed the opposing machine from the inside.  Timeframe/Other Details  I think this one dates from the 1970s.  I don't think it more recent, given my recollections of when I read it, and also the technology descriptions they used in it.  There is an outside chance that the combat is not between rival nations, but rival corporations.  That would change the premise of "man evolving beyond bloodshed", of course.  I think it's nations fighting, though.  I am almost 100% certain this was in a sci-fi magazine, not an anthology. Shuffling questions in exam Does registering [name]overflow.com violate Stack Overflow's trademark? Polynomial $p(a) = 1$, why does it have at most 2 integer roots? The question that I am trying to answer is :  Suppose is $p(x)$ is a polynomial with integer coefficients. Show that if $p(a) = 1$ for some integer a then $p(x)$ has at most two integer roots.  I have no idea how to get started. Any help would be awesome! I have a very large codebase (read: thousands of modules) that has code shared across numerous projects that all run on different operating systems with different C++ compilers. Needless to say, maintaining the build process can be quite a chore.   There are several places in the codebase where it would clean up the code substantially if only there were a way to make the pre-processor ignore certain #includes if the file didn't exist in the current folder. Does anyone know a way to achieve that?  Presently, we use an #ifdef around the #include in the shared file, with a second project-specific file that #defines whether or not the #include exists in the project. This works, but it's ugly. People often forget to properly update the definitions when they add or remove files from the project. I've contemplated writing a pre-build tool to keep this file up to date, but if there's a platform-independent way to do this with the preprocessor I'd much rather do it that way instead. Any ideas? How to call a parent class function from derived class function? Translating cooking terms between US / UK / AU / CA / NZ How can I record demos of my gameplay? I remember seeing a lot of cool Counter Strike videos back in the day. How do you go about recording your gameplay (any game) into an easy to edit format? Do you need special software? How much does it cost? Object not visible in render I have a model which you can see here (camera perspective).    In the second image, you can see how the render turned out.    I'm at my wit's end trying to figure our why it's not showing up in the render. If anyone needs more information, just ask and I'll get it to you. I've already checked the normals and they're not the cause. PXE Boot Server Installation Steps in Ubuntu Server VM Using getline(cin, s) after cin When working with radios, I've seen the term "dBm". What does this mean?  The  gives a table showing the equivalent power in Watts. Is the dBm output just the power level of a transmitter?  How does this differ from signal strength measured in dB? I need to hide field 'B' when the value in field 'A' is NOT equal to 'Other'.  This is a custom list form.  Is jquery the way to go? Segmentation fault with strcpy() How do I fill an area with faces lined up in a grid over a not flat surface? Hello to everyone again,  I'm looking for an addon or a short cut to fill fast and simmetrically a NOT FLAT surface that needs to be closed with vertices. Doing it manually it takes a lot of time. I tried to do CTRL F and then Grid fill,but it didn't work. Neither F key works. Check the attached screenshot. thanks.