Let $X_1$ and $X_2$ be real-valued independent random variables with a standard normal distribution. Let $ Y = X_1X_2$. Find the characteristic function of Y.  Attempt:  $\phi_{Y}(t)$ = $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} exp(it(x_1x_2))exp(-1/2(x_1x_2)^2)dx_1dx_2$  I know that the standard normal characteristic function is:  $\phi_{X_1}(t)$ = $exp(\frac{-t^2}{2})$ and likewise for $X_2$  However, I'm not sure what the resulting product is. I realize that I can use a Jacobian, but I'd like to keep it simple and see if there is a way to derive the characteristic function with something easier.
Can anybody please suggest how I can compute the moment generating function of the inner product of two Gaussian random vectors, each distributed as $\mathcal N(0,\sigma^2)$, independent of each other? Is there some standard result available for this? Any pointer is highly appreciated.
Rendering low and res/high res images Is it possible to render in Blender with reduced low res files and then ask Blender to render the final render with 4k high res files? Low and high res files would be numbered identically but would be located in different folders.  Many thanks Many C++ books contain example code like this...  std::cout &lt;&lt; "Test line" &lt;&lt; std::endl;   ...so I've always done that too. But I've seen a lot of code from working developers like this instead:  std::cout &lt;&lt; "Test line\n";   Is there a technical reason to prefer one over the other, or is it just a matter of coding style? How to find the 'sizeof' (a pointer pointing to an array)? First off, here is some code:  int main()  {     int days[] = {1,2,3,4,5};     int *ptr = days;     printf("%u\n", sizeof(days));     printf("%u\n", sizeof(ptr));      return 0; }   Is there a way to find out the size of the array that ptr is pointing to (instead of just giving its size, which is four bytes on a 32-bit system)? I've got a directory shared on my network, using Samba. People on the netwerk use this Samba share to download files from my share to their computers. I'm interested in what files are downloaded most, and by which users they are downloaded. I'd like to see this in real time, if possible. Is there a program that allows me to do this? If yes, how can I use it?    I'm using Ubuntu 11.10. When I am writing I often come across the words "around" and "round." I was wondering what is different between them, and how they would be used in different contexts. Do you have any useful awk and grep scripts for parsing apache logs? I can use log analyzers, but often I need to parse recent web logs to see what's happening at the moment.   I sometimes do things like to figure out top 10 ips that request a certain file  cat foo.log | grep request_to_file_foo | awk '{print $1}' |  sort -n | uniq -c | sort -rn | head   What do you have in your toolbox? Why does default auto.arima stop at (5,2,5)? Why doesn't the frequency of light change during refraction? Arch on WSL: libQt5Core.so.5 not found, despite being installed I was hit by some hardcore power-tripping forum mods on the Arch Linux forum, so here I am.  I'm trying to run GUI apps on WSL, such as dolphin or kate. However, none of them can find libQt5Core.so.5, despite it being installed in /usr/lib as you'd expect. Here's a terminal transcript:  /u/lib » DISPLAY=:0.0 dolphin dolphin: error while loading shared libraries: libQt5Core.so.5: cannot open shared object file: No such file or directory /u/lib » pwd /usr/lib /u/lib » ls -la | grep Qt5Core -rw-r--r-- 1 root root     1143 Aug  7 06:05 libQt5Core.prl lrwxrwxrwx 1 root root       20 Aug  7 06:05 libQt5Core.so -&gt; libQt5Core.so.5.11.1* lrwxrwxrwx 1 root root       20 Aug  7 06:05 libQt5Core.so.5 -&gt; libQt5Core.so.5.11.1* lrwxrwxrwx 1 root root       20 Aug  7 06:05 libQt5Core.so.5.11 -&gt; libQt5Core.so.5.11.1* -rwxr-xr-x 1 root root  5234312 Aug  7 06:05 libQt5Core.so.5.11.1* /u/lib » ldd `which dolphin`         linux-vdso.so.1 (0x00007fffdb426000) ...         libQt5Core.so.5 =&gt; not found /u/lib » pacman -Qk qt5-base qt5-base: 3546 total files, 0 missing files   Note that other GUI apps work fine - anything compiled against GTK, including Firefox, work perfectly. I just have to set the $DISPLAY variable. Not setting the var doesn't help, as you'd expect.  Any ideas? Thanks! I'm fairly new to bicycling; apologies if this is a standard question but I couldn't figure out how to phrase it to search for it elsewhere.  I ride a KHS Urban-X with Kenda Kwest tires (just what it came with). I've gotten barely any use out of it since owning it, and it's been resting comfortably in the living room through the winter, so I decided today was a great day to get back on.   So I unscrewed the back tire cap, and began to hand-pump air into the tire. All seemed to go well, and the tire held pressure fine, but when I removed the tire pump, the pin appeared to remain pushed down slightly, and air flowed back out of the tire. I looked, and this is what I saw:    Several narrow gashes that appear as if the rubbed was ripped, cut, or separated somehow.  Questions   Is this a common occurrence? Again, I've ridden on this bike very little and thus filled up the tires even less. We have no animals and no bugs around that I know of that would gnaw or chew on such material. Is this fixable, or is my tube completely shot? It would seem to me that the latter is the case. Is there anything I can do to prevent this in the future? I use a hand pump but am gentle; would a stand-up pump be a better choice?  Do bike tires come with a standard warranty for cases like this, or do you know of any brands that do? If I have to purchase new tires, I'm going to purchase Is there any kind of "off-season" maintenance that I should be performing on my bike that could prevent things like this?   Thanks in advance for any help you can give! KeyListener not working SQL Server compatibility with New TLS Standards The major browsers are moving beyond SSL3.0 and TLS1.0 . The PCI Security Council has declared an end-of-life date for these protocols to be considered sufficiently strong encryption.   We need to move away from these protocols, to use newer and stronger ones. On Windows servers, you can very easily disable these old protocols, and instead offer only TLS1.1 or greater.  However, as noted , Microsoft SQL Server 2008 R2 and SQL Server 2012 (Standard, at least) both will not start if those lower protocols are disabled.  However, there are a growing number of versions of MS SQL Server.  There are SQL Server Standard, Business Intelligence, Enterprise, Express, Web, and Compact editions.  And of course there is SQL Server 2008, 2012, 2014, and (in pre-release) 2016.  Which of these editions support or will support the use of only TLS1.1 or greater protocols? EmptyListToVisibilityConverter Re-entering the US for the third time in a year with total of 240 Days I went to the US for 90 days on an ESTA in August to November. Then back to my home country for 1 month and then back again mid December until mid February and now I am going back again after 10 days for another 90 days. Now I can see that some people refer to 180 day rule in a year. My stay will be 240 days in one year with short gaps between coming back. Is the 180 day rule accurate? Some people write that you can do unlimited trips to The US as long as you don't stay more than 90 days. Is that correct? My reason for going there the first time was to visit and travel with an American friend.  After my first visit (she is my girlfriend) moved with me to my home country but has ended up in a custody battle in the US so she needed to return urgently so I returned only to be with her and support her, not to be a tourist.  Then after my second visit I needed to go back to my sick Dad and to rent out my apartment since we left in a rush to go back to the US.  Now, my third visit is to continue to be with my girlfriend and support her in her difficult situation.   My question is, if I do end up in trouble should I tell them this whole story? Or should I just say I am visiting a friend for a third time because I want to see more of the country and I have some more time off before I go back to my job.  I have proof of money in my bank account and I stay and eat with her for free Documented proof that I have a job to return to in my home country My return ticket printed and an address to stay at in the US and I have proof I have an apartment in my home country Lack of Articles in Headlines Are there any rules as to an article before a noun in the singular in headlines please, i.e. lack of an article?       Description of Type of Investment What is the promotion rate of professors in United States? I'm writing an essay, and want to know the promotion rate and average duration from assistant professor to associate professor and associate professor to professor in recent years. Sadly I don't know the data for the US. Could anyone tell me? Typesetting long division with tabular and overlining repeating part If $E$ has measure zero, then does $E^2$ have measure zero? Currently I'm just working through measure theory just to see if its something I would like to take.  Unfortunately I am stuck on this problem from Carothers.     If $m^*(E)=0$, then $m^*(E^2)=0$.   Where $m^*$ denotes outer measure and $$E^2=\{x^2:x\in E\}.$$  I toyed with the idea that $I_k &lt; 1 \Rightarrow I^2_k &lt; I_k$.  However I am at a loss as to how to set up a chain of inequalities (which is what I am assuming I need). I'm trying to make a liquid simulation with Mantaflow where the flow of the liquid respects the flow object's local rotation, so it looks like water spraying in different directions. It looks like the options on the flow object only work for 'world' coordinates and doesn't take into account the coordinates of the object. How do I make the water 'fire' (like a hose) into a specific direction that uses local coordinates instead of world ones? How does vampirism work? A PC in our group was off on his own following a recurring villain of our groups. He was paralyzed where he could move, and the vampire bit his neck. The vampire said "now you get to remain 19 (years old) forever" before he left the cave. This was all done in RP.  We found him later, and healed up his bite marks. Our characters know of vampires, specifically only the one that bit him (recurring villain), but don't really know how vampirism works, because it isn't a super common thing in this world yet.   My question as the player is:  How does RAW vampirism work in 5e?  Is it a disease that develops over time? Is it something where they have to die and be brought back as a vampire?