What's the term for when a random variable $X$ has a higher probability of being greater than t than the probability of $Y$ being greater than $t$ for all real $t$? I recall reading a term for this somewhere but can't remember where, and this this turns out to be annoying to Google. I don't want to end up making up a new definition when one exists somewhere. Additionally, are there any other related notions of a random variable upper/lower bounding another? It doesn't really make sense to say $X &gt; Y$ for example (unless maybe if the essential min of $X$ is strictly greater than the essential max of $Y$).
I'm trying to solve an exercise about first order stochastic dominance, and there is this one question that I'm not able to answer.  Here are the hypothesis of the exercise:  Let Y ∼ F and Y' ∼ G, that means Y (resp. Y') has a probability law given by the distribution function F (resp. G).  Assume that Y ≥ 0 and Y' ≥ 0 and E(Y) + E(Y')&lt; ∞.  Here comes the question:  -Show that if the survival function of Y first order dominates over the survival function of Y', then if Y and Y' are independent, we have P(Y ≥ Y') ≥ 1/2.  I don't see how to use that Y and Y' are in this case independant.  Maybe I should set X=Y-Y' and then use the convolution?  Any help, would be appreciated, thanks !  Alex
Modular arithmetic problem (mod $13$) $$\large2017^{(2020^{2015})} \pmod{13}$$  I am practicing for my exam and I can solve almost all problem, but this type of problem is very hard to me. In this case, I have to compute this by modulo $13$. In Ubuntu 13.04 I used to use the ppa:upubuntu-com/tor64  PPA to install the 64-bit Tor Browser Bundle. However, in Ubuntu 13.10 it doesn't work anymore. How do I install the Tor Browser Bundle in Ubuntu? What really causes light/photons to appear slower in media? How does Java's serialization work and when it should be used instead of some other persistence technique? I've been lately trying to learn more and generally test Java's serialization for both work and personal projects and I must say that the more I know about it, the less I like it. This may be caused by misinformation though so that's why I'm asking these two things from you all:  1: On byte level, how does serialization know how to match serialized values with some class?  One of my problems right here is that I made a small test with ArrayList containing values "one", "two", "three". After serialization the byte array took 78 bytes which seems awfully lot for such low amount of information(19+3+3+4 bytes). Granted there's bound to be some overhead but this leads to my second question:  2: Can serialization be considered a good method for persisting objects at all? Now obviously if I'd use some homemade XML format the persistence data would be something like this  &lt;object&gt;     &lt;class="java.util.ArrayList"&gt;     &lt;!-- Object array inside Arraylist is called elementData --&gt;     &lt;field name="elementData"&gt;         &lt;value&gt;One&lt;/value&gt;         &lt;value&gt;Two&lt;/value&gt;         &lt;value&gt;Three&lt;/value&gt;     &lt;/field&gt; &lt;/object&gt;   which, like XML in general, is a bit bloated and takes 138 bytes(without whitespaces, that is). The same in JSON could be   {     "java.util.ArrayList": {         "elementData": [             "one",             "two",             "three"         ]     } }   which is 75 bytes so already slightly  smaller than Java's serialization. With these text-based formats it's of course obvious that there has to be a way to represent your basic data as text, numbers or any combination of both.  So to recap, how does serialization work on byte/bit level, when it should be used and when it shouldn't be used and what are real benefits of serialization besides that it comes standard in Java? Continuity of $L^1$ functions with respect to translation Reinstall software-center in Ubuntu 16.04 Bounded sequence with divergent Cesaro means Is there a bounded real-valued sequence with divergent Cesaro means (i.e. not Cesaro summable)?  More specifically, is there a bounded sequence $\{w_k\}\in l^\infty$ such that $$\lim_{M\rightarrow\infty} \frac{\sum_{k=1}^M w_k}{M}$$ does not exist?  I encountered this problem while studying the limit of average payoffs criterion for ranking payoff sequences in infinitely repeated games. RECENTLY ON THE ANDROID SITE WE GOT  WRITTEN IN ALL CAPS.  THIS WAS SURPRISING TO ME, SINCE I ASSUMED IT WOULD HAVE BEEN CAUGHT BY THE QUALITY FILTER &mdash; BESIDES BEING INDICATIVE OF OTHER POST QUALITY ISSUES, IT'S AWFUL TO LOOK AT AND REQUIRES A LARGE AMOUNT OF EDITING TO FIX (EVEN IF YOU RUN IT THROUGH TOLOWER() FIRST).  WE ALL KNOW THAT CAPS LOCK IS , BUT STILL: CAN WE BLOCK THESE QUESTIONS?    RICK  RICK  OH MAN THAT STUFF JACOB G  IS COMEDY GOLD  RIIICK     Note to editors: This post looks exactly like it is supposed to look, as an example of a question in all-caps. Please do not replace the uppercase letters with lowercase. I am having trouble with the following problem.      Let $R$ be an integral domain, and let $a \in R$ be a non-zero element. Let $D = \{1, a, a^2, ...\}$. I need to show that $R_D \cong R[x]/(ax-1)$.    I just want a hint.  Basically, I've been looking for a surjective homomorphism from $R[x]$ to $R_D$, but everything I've tried has failed. I think the fact that $f(a)$ is a unit, where $f$ is our mapping, is relevant, but I'm not sure. Thanks Why must fermions be antisymmetric? I have read that fermions cannot exist in the same state simultaneously. I understand why indistinguishable particles with an antisymmetric superposition of states can't exist in the same state simultaneously, but why must fermions have an antisymmetric superposition of states?  The only characterising property I know of fermions having besides antisymmetry is spin, for which they have half-integer. I understand that this is the case simply because particles with half-integer spin and particles with zero or integer spin were defined as fermions and bosons respectively.  My perusal of the Wikipedia page on the  leaves me under the impression that spin has nothing to do with the wavefunction's symmetry properties:     Naively, neither has anything to do with the spin, which determines the rotation properties of the particles, not the exchange properties.   Are antisymmetric wavefunctions simply classified as fermions, in the way half-integer spin particles were? I don't see how this could be the case, as, if spin and symmetry were independent, half-integer spin particles with symmetric wavefunctions (and antisymmetric integer spin particles) would be possible. I'm looking to take a spontaneous trip for a long weekend. The date is flexible and the destination can be anywhere in the world. Is there a site that would allow me to search airfares within a date range, while keeping the destination parameters open? Why does reversing serial upvoting not reverse the Mortarboard badge? user has serial up-voting reversed, but he still have badge Mortarboard.   Is it correct behavior of badges? I suppose it should be reversed too.  And will this 200 daily reputation (2012-03-20) score for the Epic, and Legendary badges? How to change keyboard layout in Lubuntu 15.10? I have recently upgraded my PC from Lubuntu 14.10 to 15.10. Before the upgrade my keyboard layout matched my Danish keyboard. But after the upgrade I probably have a standard English/US layout.   Preferences/Language Support, doesn't let me configure the keyboard layout. Preferences/Keyboard and Mouse, only let me configure stroking delay and similar.    I don't know about iBus and fcitx, as far as the tooltip infomation tells me, its for more complex languages such as Chinese.  I don't have a US icon in the taskbar, no keyboard and/or language icon at all.  Things I have tried:   Running the following in a terminal works, but only until the next reboot:  setxkbmap -layout dk  I got the following parameter in the file /etc/default/keyboard:  XKBLAYOUT="dk"  Installing and running the app Lxkeymap changes the keyboard to Danish when I run it, but rebooting will change the layout back to US.   I don't want anything fancy, I just want to set my keyboard layout to Danish. How can I do that? Best practice: card expiry date on checkout How are answers with the same vote score sorted? proof relating to limit definition of e I'm studying the book A Primer for the Mathematics of FinancialEngineering, along with it's solution guide. I've ran into trouble with the solution to question 7 in chapter 1.    The problem is:  Show that $(1+\frac{1}{x})^x &lt; e &lt; (1+\frac{1}{x})^{x+1}$ .   Specifically, I'm having trouble with one step in the solution.    It says that  $$x\ln\left(1+\frac{1}{x}\right)&lt;1&lt;(x+1)\ln\left(1+\frac{1}{x}\right)$$  can be written as   $$\frac{1}{x+1} &lt;\ln\left(1+\frac{1}{x}\right)&lt;\frac{1}{x}, \forall \ge 1.$$  How can the first equality (or is it inequality?) be written as the second? How can I check for Python version in a program that uses new language features? If I have a Python script that requires at least a particular version of Python, what is the correct way to fail gracefully when an earlier version of Python is used to launch the script?  How do I get control early enough to issue an error message and exit?  For example, I have a program that uses the ternery operator (new in 2.5) and "with" blocks (new in 2.6).  I wrote a simple little interpreter-version checker routine which is the first thing the script would call ... except it doesn't get that far. Instead, the script fails during python compilation, before my routines are even called. Thus the user of the script sees some very obscure synax error tracebacks - which pretty much require an expert to deduce that it is simply the case of running the wrong version of Python.  I know how to check the version of Python. The issue is that some syntax is illegal in older versions of Python. Consider this program:  import sys if sys.version_info &lt; (2, 4):     raise "must use python 2.5 or greater" else:     # syntax error in 2.4, ok in 2.5     x = 1 if True else 2     print x   When run under 2.4, I want this result  $ ~/bin/python2.4 tern.py  must use python 2.5 or greater   and not this result:  $ ~/bin/python2.4 tern.py    File "tern.py", line 5     x = 1 if True else 2            ^ SyntaxError: invalid syntax   (Channeling for a coworker.) What would happen if I used regeneration on a creature that was the target of Annihilating Fire? For example, what would happen if a freshly cast  were to be regenerated in response to ?  My hypothesis is that it depends on whose turn it is. The regeneration being essentially a replacement effect, when two replacement effects were to be applied simultaneously, they are to be chosen in APNAP order. Is this correct?  I haven't thought this through to the end, though, so I'm not sure what would happen in the end. Adding items to page template dropdown on Page Edit Screen I am developing a WP theme with MVC approach. It only have index.php, functions.php and styles.css on the parent directory. So, I do not want to place page templates on it rather then I want to programmatically provide them from View classes while functionality from edit screen stays the same.   Users need to have page templates select option on the edit screen. How do I successfully add Items to the template dropdown?  I tried to hook into  filter.   Example code:  add_filter( 'theme_page_templates', function($templates){      $templates['my-page-template.php'] = "My Page Template";      return $templates;  });   This does not work because of the use of  on the filtered array, which removes the added page template item. I don't understand why this function is used. It seems you can only remove page template from the list but can not add a new one using the given filter.  Is there any other way around? Inequalities for combinations of $\int f $ and $\int (1/f)$ where $m\le f\le M$ on an interval