It seems like different books define continuous random variables differently.    pdf definition: The random variable X is continuous if a nonnegative function f exists, that is defined for all $x \in (-\infty, \infty)$ such that $P(X \in B) = \int_B f(x) dx$ (Sheldon Ross) cdf definition: The random variable X is continuous if its cdf is continuous and differentiable. Moreover, the derivative of the cdf is continuous except for a finite number of points. (Wackerly) Another cdf defintion: The random variable X is continuous if the cdf is a continuous function. (Wikipedia ) uncountable range:  The random variable X is continuous if its range is uncountable infinite/set of possible values is uncountable infinite. uncountable sample space:  The random variable X is continuous if the sample space is uncountable infinite.   My question are:   a) How do these definitions relate to each other? Are they equivalent? Why?  b) What is the difference between a continuous and an absolute continuous random variable?  c) Is it possible that the difference between continuous and discrete random variables vanishes from the point of view of measure theory? I have looked into several books about probability based on a rigid measure theoretic approach and none of them even mention the fact that there are different kinds of random variables.  d) In particular, is 4.) and 5.) equivalent? To me this doesnt seem to be necessarily the case...  e) Which is the "correct" definition from the above for a continuous random variable?  I would be happy also about answers from the point of view of measure theory, which probably answers this questions the most precisely.
In the book Limit Theorems of Probability Theory by Valentin V. Petrov, I saw a distinction between the definitions of a distribution being &quot;continuous&quot; and &quot;absolutely continuous&quot;, which is stated as follows:  &quot;...The distribution of the random variable $X$ is said to be continuous if $P\left(X \in B\right)=0$ for any finite or countable set $B$ of points of the real line. It is said to be absolutely continuous if $P\left(X \in B\right)=0$ for all Borel sets $B$ of Lebesgue measure zero...&quot;  The concept I am familiar with is:  &quot;If a random variable has a continuous Cumulative Distribution Function, then it is absolutely continuous.&quot;  My questions are: are the two descriptions about &quot;absolute continuity&quot; in (1) and (2) talking about the same thing? If yes, how can I translate one explanation into the other one?
$G$ is the set of all subsets of a set $A$, under the operation of $\triangle\;$: Symmetric Difference of sets.  $A$ has at least two different elements.  I need to check if this is a group, and if it does to show if the group is abelian and/or finite.  Associativity - easy from Symmetric Difference. Identity element - empty group. Inverse element - each element is inverse to itself. Am I right?  abelian? finite? Changing default width of blocks in beamer I use a custom Beamer theme for my presentations. In my presentations I'd like to use blocks (\begin{block}…\end{block}). These blocks are rendered slightly too wide to fit into my theme. Is there a way how I can set the default width of blocks, e.g. to .9\textwidth? What is the command to truncate a SQL Server log file? I need to empty an LDF file before sending to a colleague.  How do I force SQL Server to truncate the log? MWE:  \documentclass[a4paper,12pt,twoside]{memoir} \usepackage{DejaVuSerifCondensed} \usepackage[T1]{fontenc} \begin{document} \noindent\scshape{Hello World}\\ \textsc{Hello World} \end{document}   Explanation and question:  As you can see, I am working with a memoir document class and I've load a font DejaVuSerifCondensed, I am trying to obtain small caps for my header (which I did not place in the MWE as unnecessary). However I can't obtain the desire effect with the \scshape command nor with the \textsc{}.  I've seen other questions like     .   But either they do not adapt to my problem (for what I can see) or they do not solve the problem (following the instructions there and "inserting" the piece of code in mine).  Question here is: is there a general method (piece of code or something) to obtain small caps from a custom font (even if that is one I upload with a ttf)?, if not, what would be the piece of code to select the standard small caps font? Create a shortcut for URL? How to disable touchscreen permanently on ubuntu 18.04 Metropolis is not working I tried to use the metropolis beamer theme, but I got the following error  ! Package beamerfontthememetropolis Error: Patching section title failed. See the beamerfontthememetropolis package documentation for explanation. Type H &lt;return&gt; for immediate help.... \patchcmd Distance between surfaces How to find out the default file manager in the terminal? I need a way of encoding the geometry line symbol. $\overline{AB}$ gives me the line segment symbol and $\overrightarrow{AB}$ gives me the ray symbol, I need a way to get a bar over the letters that has arrows on both ends. $\overrightleftarrow{AB}$ does not work.   Thanks! I flagged a question, , as "low quality" but a moderator declined it. I thought that the post was a low quality one and would like to know why the moderator did not.  Hopefully shedding light on this matter will avail me of the knowledge to be a better flagger in future.  My logic was that mysql-fetch-row is not supposed to return an array of results. If it does than its name would be mysql-fetch-rows.  What's your take on the situation? Do subdomains help/hurt SEO? If I have example.com, and blog.example.com, do search engines see those as one site or two? Will using subdomains hurt SEO? I have some technical reasons why I can't use example.com/blog. Feed can show entries for deleted posts, which viewing them fails Just noticed this:  Someone @replied to me in a question which appropriately showed up in my feed.  I go to click on that notice in the feed, and it presents me with "Loading the question failed."  I'd show a picture, but I don't think that it's entirely relevant - I believe it's reproducible if an answer is posted, someone makes a reply, the answer is removed, then that someone else goes back in the feed to try and revisit the post.  Alternatively, why not synchronize the interface between what a 10K user sees on the site, and what they see in the app?  As a 10K user, I would expect to see deleted questions and answers. Dr. StrangeNumbers or: How I Learned to Stop Worrying and Love Floating Point Arithmetic This is a repost with additional info of an older question that was never answered. I have stumbled upon this old question while also trying to find the name of the book series myself, as I dearly wish to read it again but for the life of me, I can't remember the name of it.      This is a long shot but I'm really in need of help for this because      it was an amazing book. It was either a kid's or YA book. A guess      from what year it was published would be 90s-mid 2000s, as I read it      around 2008-2011.         The book cover was in first person view, with the main character   holding a weapon, and was a 3D rendered image inside a dungeon or   cave (that was very reminiscent of early 90s CGI animation), though   there may be variants.    It would have been set around when computer   games were mainstream (maybe around the same time as when it might've   been published) It's about a teenage boy who is bored at home, so he   looks through an old family computer to play a game and finds one   that he has never seen before. When he plays it he gets transported   into the game which is a fantasy-medieval setting.    All that I can   remember about the game and story is that the main character is on a   quest, he rides horses a lot, (to the point where he can ride horses   perfectly in real life), at one point he eats food at an ingame   tavern/restaurant? and he tosses the ingame coin currency at the   server because he's in a hurry, (don't know why I remember that bit   specifically haha), and he meets a gnome/dwarf character who is   helping him on his journey, who has a home in an ingame mountain, and   that time goes faster in the game. (For example, if he spent 5   minutes in the real world, 3 in-game days would have passed)   Sometimes as he's playing something stops him from being able the   play the game for a few weeks, maybe a concerned parent because he's   playing too much or there's a family vacation. I remember during the   time when he's not playing the game he's on a farm and knew how to   ride the horses there.    There was a character in the real world who   was the main character's sibling, relative or friend. They were like   a rival and was also very whiny (to some extent). They were on the   farm as well with him and questioned how the main character knew how   to ride a horse when he had never done it in real life.   This is all I can remember but if you need more specific information I can try and answer. It is definitely not Space Demons or Demons Don't Dream. If anyone knows anything remotely similar please mention it, I would really appreciate it. Thank you!!            Additional info:    It is not an online game or virtual reality game. It was a normal PC game, and all other characters are (I think), just NPCs.  He woke up for the first time in a field/farm and was super confused about what happened and where he was, as he had never played the game before. He is asked a super serious question before starting the game up for the first time, like "are you sure you wish to play ___?", and then he is transported into the game.  He frequently exited the game, to continue his IRL life. And sometimes, he would lose access to the family desktop PC for a couple of days at a time, and when he went back everyone in the game (NPCs), thought he had disappeared because he was gone for ages. I think once, his family when on holidays and he couldn't take the PC, for example.  It was a book series, not a single book. In later books he would play the game again months later in the real world, and years and years would have passed in the game world.  The plot involved 2 main warring sides/factions/countries of humans, but there were fantasy/magical races and characters involved. I vaguely remember something about each side having official colours, one red and one blue. I think the main character was on the blue side.   The original question was never correctly answered, so I'm posting again to see if anyone can remember.   Books that were ruled out in original post: Eden's Gate &amp; Heir Apparent. Once and for all: Is there anything that can possibly be done to my commander that would cause it to cease being my commander?    All of these questions ask the same thing, but aren't general enough to be marked as a duplicate.      Is a commander still a commander if it gets turned into a forest?     Is a commander still a commander if it gets turned face down? I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$ How does IPv4 Subnetting Work? This is a  about IPv4 Subnets.      Related:               How does Subnetting Work, and How do you do it by hand or in your head?  Can someone explain both conceptually and with several examples? Server Fault gets lots of subnetting homework questions, so we could use an answer to point them to on Server Fault itself.   If I have a network, how do I figure out how to split it up? If I am given a netmask, how do I know what the network Range is for it? Sometimes there is a slash followed by a number, what is that number? Sometimes there is a subnet mask, but also a wildcard mask, they seem like the same thing but they are different? Someone mentioned something about knowing binary for this? Virtualbox has no 64-bit options What is the Unix command to create a hardlink to a directory in OS X? How do you create a hardlink (as opposed to a symlink or a Mac OS alias) in OS X that points to a directory? I already know the command "ln target destination" but that only works when the target is a file. I know that Mac OS, unlike other Unix environments, does allow hardlinking to folders (this is used for Time Machine, for example) but I don't know how to do it myself.