I'm having a hard time to do this exercise. I'm using the definition for convolution but I'm stuck at the integral. Thanks in advance.  For $t&gt;0$ consider $f_a(x)=\frac{1}{\sqrt{4 \pi a}}e^{-\frac{x^2}{4a}}$. Show that $f_a*f_b=f_{a+b}$  By definition: $(f*g)(x)$= $\int_{-\infty}^{+\infty} f(u)g(x-u) du$ One computes, $(f_a*f_b)(x)=$ $\int_{-\infty}^{+\infty} \frac{1}{\sqrt{4 \pi a}}e^{-\frac{u^2}{4a}} \frac{1}{\sqrt{4 \pi b}}e^{-\frac{(x-u)^2}{4b}} du$ $=\frac{1}{\sqrt{4 \pi a}}\frac{1}{\sqrt{4 \pi b}} \int_{-\infty}^{+\infty} e^{-\frac{u^2}{4a}}e^{-\frac{(x-u)^2}{4b}}$ I can't solve this integral. I tried using Gaussian Integral but still cant solve it.
I know that the product of two Gaussians is a Gaussian, and I know that the convolution of two Gaussians is also a Gaussian. I guess I was just wondering if there's a proof out there to show that the convolution of two Gaussians is a Gaussian.
How to completely uninstall and remove QGIS 2.8.1 on Windows Vista? I want to uninstall and completely remove QGIS installation from my computer(windows vista). I have uninstalled it and when i try to reinstall it says it is already installed on the system. Is PCA still done via the eigendecomposition of the covariance matrix when dimensionality is larger than the number of observations? I have a $20\times100$ matrix $X$, containing my $N=20$ samples in the $D=100$-dimensional space. I now wish to code up my own principal component analysis (PCA) in Matlab. I demean $X$ to $X_0$ first.  I read from someone's code that in such scenarios where we have more dimensions than observations, we no longer eigen-decompose the $X_0$'s covariance matrix. Instead, we eigen-decompose $\frac{1}{N-1}X_0X_0^T$. Why is it correct?  The normal covariance matrix is of size $D\times D$, each element of which tells us the covariance between two dimensions. To me, $\frac{1}{N-1}X_0X_0^T$ is not even of the correct dimensions! It is $N\times N$ matrix, so what would it tell us? Covariance between two observations?! Cancelling upvote on comment? As mentioned on :     For something that can be done so   easily by accident, there should be   the ability to undo mistakes.   I know that I have clicked several times on a comment up-vote without actually wanting to.  I would suggest being able to cancel the up-vote (maybe only for a couple of minutes or even less than a minute, no problem, as long as I can undo a miss-click). Pacf lag axe , is not an integer I'm trying to use ARIMA process to predict the behaviour of a time series, the probleme I face is that I can't get the order of each component of ARIMA, the lag is between 0 and 1, same goes for the acf. The time series has no tendency, and I guess no seasonality.     My question is : Does this mean that ARIMA orders are all 0s, and if so, is the HoltWinters a good (or the best) solution for modelling this kind of TS ? Prove that $f$ has a fixed point . Sort arrays of primitive types in descending order I've got a large array of primitive types (double). How do I sort the elements in descending order?  Unfortunately the Java API doesn't support sorting of primitive types with a Comparator.  The first approach that probably comes to mind is to convert it to a list of objects (boxing):  double[] array = new double[1048576];     Arrays.stream(array).boxed().sorted(Collections.reverseOrder())…   However, boxing each primitive in the array is too slow and causes a lot of GC pressure!  Another approach would be to sort and then reverse:  double[] array = new double[1048576]; ... Arrays.sort(array); // reverse the array for (int i = 0; i &lt; array.length / 2; i++) {      // swap the elements      double temp = array[i];      array[i] = array[array.length - (i + 1)];      array[array.length - (i + 1)] = temp; }   This approach is also slow - particularly if the array is already sorted quite well.  What's a better alternative? Site.createPortalUser() returning null in test context Where should the comma be placed in the salutation of a letter? Sometimes I see a comma after the proper name:     Hello Mr. Black,      In order to give you....   But my native language is not English and I think that the comma in this phrase should be placed before the proper name:      Hello, Mr. Black.      In order to give you....   What is the correct punctuation in English? If $X,Y$ are equivalence relations, so is $X \times Y$ Badge overflow on users with too many badges When a user has &quot;too much&quot; reputation and &quot;too many&quot; badges, the design of the user card breaks:    The badge counters overflowing to another line. Can anything be done about this? For example, expanding the rep and badges to their own lines:  NameRepBadges I'm struggling with some concepts in my algebra class. I've chosen these two completely unrelated examples that demonstrate my struggling topics (characteristics and function rings) the two are the following:  $1$) Let $X$ be a set, and let $S$ be a ring. Give a proof that the left distributive law holds for $(S^X, +, *)$.  $2$) Suppose $Y^3 = Y$ for each element $Y$ of a unital ring $R$. Show that $R$ has a finite characteristic d that is a divisor of $6$.  Any example solutions for these proofs would be helpful. Thank you. How can I modify the advanced search form when my hook runs too early? How exactly does curved space-time describe the force of gravity? I understand that people explain (in layman's terms at least) that the presence of mass "warps" space-time geometry, and this causes gravity. I have also of course heard the analogy of a blanket or trampoline bending under an object, which causes other objects to come together, but I always thought this was a hopelessly circular explanation because the blanket only bends because of "real" gravity pulling the object down and then pulling the other objects down the sloped blanket. In other words, to me, it seems that curved space wouldn't have any actual effect on objects unless there's already another force present.   So how is curved space-time itself actually capable of exerting a force (without some source of a fourth-dimensional force)?  I apologize for my ignorance in advance, and a purely mathematical explanation will probably go over my head, but if it's required I'll do my best to understand. How do I install a different Python version using apt-get? QR code scanner I have looked at other questions on this site (e.g. "why does space expansion affect matter") but can't find the answer I am looking for.  So here is my question: One often hears talk of space expanding when we talk about the speed of galaxies relative to ours. Why, if space is expanding, does matter not also expand? If a circle is drawn on balloon (2d plane), and the balloon expands, then the circle also expands. If matter is an object with 3 spatial dimensions, then when those 3 dimensions expand, so should the object.  If that was the case, we wouldn't see the universe as expanding at all, because we would be expanding (spatially) with it.  I have a few potential answers for this, which raise their own problems:   Fundamental particles are 'point sized' objects. They cannot expand because they do not have spatial dimension to begin with. The problem with this is that while the particles would not expand, the space between them would, leading to a point where the 3 non-gravity forces would no longer hold matter together due to distance Fundamental particles are curled up in additional dimensions a la string theory. These dimensions are not expanding. Same problems as 1, with the added problem of being a bit unsatisfying. The answer seems to be (from Marek in the previous question) that the gravitational force is so much weaker than the other forces that large (macro) objects move apart, but small (micro) objects stay together. However, this simple explanation seems to imply that expansion of space is a 'force' that can be overcome by a greater one. That doesn't sound right to me. I need help with STATS homework about normal distribution The distribution of sale price for new MacBook Air’s is approximately normally distributed with a mean of \$800 and a standard deviation of \$70. (a) Mr. Bob plans to spend between $700 \text{ and } $850 on a new Apple MacBook Air. What proportion of the available MacBook Airs can Mr. Bob afford? (b) What is the 90th percentile of computers of this type? (c) Show that a MacBook Air priced at \$1000 would be considered an outlier using the 1.5 × IQR rule. For a, I tried doing 850-700=50 and $\frac{50}{800} = \frac{1}{40}$, for b, I tried $m+(0.5 \times R)/y$ for c, I tried $1.5 \times 700 - 1.5 \times 850$ Fibonacci Recurrence Relations Solve the recurrence relation $f(n) = f(n − 1) + f(n − 2)$ with initial conditions $f(0)=1$, $f(1)=2.$   So I understand that it grows exponentially so $f(n) = r^n$ for some fixed $r$.  This means substituting this $r^n = r^n-1 + r^n-2$ which gives the characteristic equation of $r ^ 2 - r - 1 = 0$. I'm not 100% sure where to move on from here. How do you get a submit button in the Windows Phone keyboard? Generating list of manually installed packages and querying individual packages I'd like to get a list of packages installed manually by apt or aptitude and be able to find out whether a foobar package was installed manually or automatically. Is there any neat way of doing that from the command line?