Understanding Countability in Sample Spaces In [Casella, Berger] Statistical Inference there is a short discussion on countability of sample spaces and its implications:  This distinction between countable and uncountable sample spaces is important only in that it dictates the way in which probabilities can be assigned. For the most part, this causes no problems, although the mathematical treatment of the situations is different. On a philosophical level, it might be argued that there can only be countable sample spaces, since measurements cannot be made with infinite accuracy. (A sample space consisting of, say, all ten-digit numbers is a countable sample space). While in practice this is true, probabilistic and statistical methods associated with uncountable sample spaces are, in general, less cumbersome than those for countable sample spaces, and provide a close approximation to the true (countable) situation.  I'm having a bit of trouble wrapping my head around the nuances of the distinction, and concretely picturing the implications discussed. A couple of questions: how is the way that probabilities are assigned different in an uncountable space? Can you provide an example of the way probabilities are assigned to some uncountable space and how it differs from assignment to the countable space that it is closely approximating? Further, can you provide an example of how the mathematical treatments are different? What makes the uncountable spaces less cumbersome to work with (I would imagine it would be the opposite)? Lastly, I'm not too sure what they mean by measurements of infinite accuracy in this context. While of course the finite set of ten-digit numbers is of finite accuracy and we can certainly work with it, why would a countably infinite be any more philosophically sound to work with? Wouldn't it also be of infinite accuracy (I'm thinking of e.g. the natural numbers, although again maybe I am misunderstanding what is meant by accuracy)? I imagine some of these could be answered as I read on, but at the very least I'd like to have something in my mind so I know what to pay attention to as I keep reading. I also haven't been able to find much other discussion online regarding the nuances of countability, so any insights or references that I missed would be greatly appreciated.
Why do we need sigma-algebras to define probability spaces? We have a random experiment with different outcomes forming the sample space $\Omega,$ on which we look with interest at certain patterns, called events $\mathscr{F}.$ Sigma-algebras (or sigma-fields) are made up of events to which a probability measure $\mathbb{P}$ can be assigned. Certain properties are fulfilled, including the inclusion of the null set $\varnothing$ and the entire sample space, and an algebra that describes unions and intersections with Venn diagrams.  Probability is defined as a function between the $\sigma$-algebra and the interval $[0,1]$.  Altogether, the triple $(\Omega, \mathscr{F}, \mathbb{P})$ forms a probability space.  Could someone explain in plain English why the probability edifice would collapse if we didn't have a $\sigma$-algebra? They are just wedged in the middle with that impossibly calligraphic "F". I trust they are necessary; I see that an event is different from an outcome, but what would go awry without a $\sigma$-algebras?     The question is: In what type of probability problems the definition of a probability space including a $\sigma$-algebra becomes a necessity?       provides a plain English accessible explanation. The idea is a spinning pointer rotating counterclockwise on a circle of unit perimeter:       We begin by constructing a spinner, which consists of a circle of unit   circumference and a pointer as shown in [the] Figure. We pick a point   on the circle and label it $0$, and then label every other point on the   circle with the distance, say $x$, from $0$ to that point, measured   counterclockwise. The  experiment  consists  of spinning the pointer and    recording the label of the point at the tip of the pointer. We let the random   variable $X$ denote the value of this outcome.  The sample space is   clearly the interval $[0,1)$.  We would like to construct a   probability model in which each outcome is equally likely to occur. If   we proceed as we did [...] for experiments with a finite number of   possible outcomes,  then we must assign the probability $0$ to each   outcome,  since otherwise,  the  sum  of  the  probabilities,  over    all  of  the  possible  outcomes,  would not equal 1.  (In fact,   summing an uncountable number of real numbers is a tricky business;    in  particular,  in  order  for  such  a  sum  to  have  any  meaning,   at  most countably many of the summands can be different than $0$.)    However, if all of the assigned probabilities are $0$, then the sum is   $0$, not $1$, as it should be.   So if we assigned to each point any probability, and given that there is an (uncountably) infinity number of points, their sum would add up to $&gt; 1$.
Does $\int_{ - 1}^1 {\frac{1}{x}dx} = 0 \text { or } \infty $? How much Valyrian steel is there that is unused for link making and why aren't people making weapons out of it? I was reading  that stated Maesters can have Valyrian steel links:     Valyrian Steel links (for the study of Magic + the Occult) are exceedingly rare and held by less than one Maester in a hundred. Ironically, they aren't seen as especially prestigious but rather viewed with suspicion.   This made me wonder how much Valyrian steel there is lying around for Maesters to be adding them to their chains. It seems like such  resource like this would be used for crafting weapons not being kept as ingots ready for Maesters to make links.  Indeed the Wiki I linked above claims:     the amount of Valyrian steel in the world is finite and extremely rare   How much of his extremely rare resource is being kept in reserve for link making and why hasn't it been used for more useful stuff like making weapons? System of linear equations having a real solution has also a rational solution. I saw this question     Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $b ∈ \mathbb{Q}^m$. Suppose   that the system of linear equations $Ax = b$ has a solution in   $\mathbb{R}^n$. Does it necessarily have a solution in $\mathbb{Q}^n$?   and I thought I'd give an interesting, possibly wrong, approach to solving it. I'm not sure if such things can be done, if not maybe you can help me refine.    I considered the form of the equality as  $$ A^{(1)}x_1+\cdots+A^{(n)}x_n=b, $$  where $A^{(i)}$ is a column vector of $A$. I then noticed that for $x_i\in\mathbb{R\setminus Q}=\mathbb{T}$ then, and this is where I think I'm doing something forbidden, each $x$ has the represenation  $$ x_1  =  k_{11}\tau_1+\cdots+k_{1p}\tau_p \\  x_2= k_{21}\tau_1+\cdots+k_{2p}\tau_p \\ \vdots \hspace{4cm} \vdots \\ x_n=k_{n1}\tau_1+\cdots+k_{np}\tau_p,$$  where $\tau_i$ is a distinct irrational number, $k_{ij}\in\mathbb{R}$, and $p$ is the number of such distinct irrational numbers. I wound this out, but there may be a discrepancy with $p$ and $m$. I feel this method can lead me to the answer, but I'm not sure where to go from here.  EDIT$^1$:  I end up getting something like this, I believe, after substitution:  $$ A(k^{(1)}\tau_1+\cdots+k^{(p)}\tau_p)=b$$  Here, $k^{(i)}$ is the vector  $$\begin{pmatrix} k_{1i} \\ \vdots \\ k_{ni} \end{pmatrix}.$$  EDIT$^2$:  I think there is no discrepancy with $p$ and $m$ because $A\in M_{m\times n}(\mathbb{Q})$, $K\in M_{n\times p}(\mathbb{R})$, and $\tau\in M_{p\times 1}(\mathbb{T})$, so  $$ (m\times n)\cdot (n\times p)\cdot (p\times 1) = m\times 1. $$ 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 pre-install Ubuntu for someone (OEM install)? I would like to pre-install Ubuntu on a computer for someone. I may want to customize it for them, but in the end they need to be able to choose a username, password, etc. when they first boot it up. (Also called an OEM install).  How do I do this? If parsing the output of ls is dangerous because it can break on some funky characters (spaces, \n, ... ), what's the best way to know the number of files in a directory?  I usualy rely on find to avoid this parsing, but similarly, find mydir | wc -l will break for the same reasons.  I'm working on Solaris right now, but I'm looking for a answer as portable across different unices and different shells as possible. Convert NSDate to NSString In Cycles, is there a way to create a procedural texture that follows the longitudinal lines of a sphere? Like a beach ball or globe? The texture should be able to be mapped to any sphere-shaped object, not just perfect spheres.    My current approach is to use a rotated wave texture, but that doesn't come from the poles (It uses latitudes) Given that every class gets a minimum of 5 ability score improvements each at levels 4, 8, 12, 16, and 19 (Rogue and Fighter get more at other levels), do these ability score improvements occur by character level or by class level?  For example, is it possible to create a character who never once gains a single ability score improvement by reaching level 3 in a class and then switching to a different class repeatedly?  I had originally thought the 5 that every class gets would occur by character level, so for example a sorcerer1/bard3 would earn an ability score improvement for reaching character level 4. But there are others who have expressed the opinion that this is wrong. I learned that the force from a magnetic field does no work. However I was wondering how magnets can be used to pick up pieces of metal like small paperclips and stuff. I also was wondering how magnets can stick to pieces of metal like a refrigerator? Intuitive explanation of the bias-variance tradeoff? I am looking for an intuitive explanation of the bias-variance tradeoff, both in general and specifically in the context of linear regression. Why doesn't grep using pipe work here? I have the following command:  find / -name libGL.so.1   Which returns lots of lines with "Permission denied". I want to exclude such lines, so I added the following:  find / -name libGL.so.1 | grep -v 'denied'   But the output is the same - my grep -v 'denied' is not filtering out the lines with Permission denied. I've tried many variations, looked over grep tutorials, but I cannot figure out the problem. Any suggestions? PHP is not working well on Ubuntu 13.10 and mcrypt is missing in phpmyadmin I've upgraded from UBUNTU 13.04 to 13.10 but I can not work with PHP pages or phpmyadmin . I've tried this way to install lamp on Ubuntu sudo apt-get install lamp-server^ phpmyadmin and I've done all of the configuration correctly after installation I've added this line  Include /etc/phpmyadmin/apache.conf to  /etc/apache2/apache2.conf then I restarted apache2   Now I have two problems:   In phpmyadmin on the bottom of the page is this error : The mcrypt extension is missing. Please check your PHP configuration  I've check  and mcrypt was in it , but in phpmyadmin it gives me error of missing .   The other problem is on PHP pages it seems like there is no PHP and it's all html because lots of PHP lines are printed in textbox's like : &lt;? echo $row['details']; ?&gt; Can anybody tell me what should I do ? Syntax to transfer from remote to local scp by leapfrogging onto another host? How to draw a torus Is there an easy way to draw a contour image of torus below with tikz? Or for that matter with any other graphics package. I have for quite some time now tried to find a satisfactory answer to this, but I haven't yet. In perturbation theory, with small parameter $\lambda$, we expand the eigenstate as  $$| E \rangle=| E^{(0)} \rangle + \lambda | b \rangle + ...$$  Where $| E^{(0)} \rangle$ is an eigenstate of the unperturbed Hamiltonian. The problem is that when there is degeneracy, there is a choice of these eigenstates. I know the answer is that they are chosen to be eigenvectors of the perturbing Hamiltonian, but my question is why. How to remove duplicate entries in dash? How to use unicode characters in Windows command line? How can I cite a range of papers, the output is a range of numbers, instead of a list of numbers? In other words, when I type  ... some dummy text here is due to me \cite{me1, me2, me3, me4} ...   I want the output to be     ... some dummy text here is due to me [3-6] ...   instead of     ... some dummy text here is due to me [3.4.5.6] ...   (I use LaTeX, and am currently using the amsart class and whatever it includes. But I am open to other suggestions.) If $F:M\to N$ is a smooth embedding, then so is $dF:TM\to TN$.