What is the best book to learn probability - Poisson, binomial, regression, etc.  I am working as an odds adjuster at a bookmaker and need to advance my skills to an odds compiler level.
I'm studying machine learning and every book I open I bump into chi-squared distribution, gamma-function, t-distribution, Gaussian, etc.   Every book I have opened so far only defines what the distributions are: they don't explain or give the intuition on where the specific formulas for the functions come from.   For example, why is chi-squared distribution the way it is? What is the t-distribution? What is the intuition behind the distribution? Proofs? etc.   I would like to have a clear and fundamental understanding of the most commonly used distributions so that every time later on when I see them, I truly understand what is a t-distribution, what is a Gaussian distribution and most importantly why are they the way they are.   It would be nice if the books / tutorials  can explain the concepts to a layman so that in order to understand them you don't already need to understand them x) Many books are like this, they don't fit for beginners :(
I have a  to which I'd like to add JPEG and PNG images that I generate on the fly.  All the examples I've seen so far in the , specially in the  use ImageIcons.  I'm generating these images as byte arrays, and they are usually larger than the common icon they use in the examples, at 640x480.   Is there any (performance or other) problem in using the ImageIcon class to display an image that size in a JPanel? What's the usual way of doing it? How to add an image to a JPanel without using the ImageIcon class?   Edit: A more careful examination of the tutorials and the API shows that you cannot add an ImageIcon directly to a JPanel. Instead, they achieve the same effect by setting the image as an icon of a JLabel. This just doesn't feel right... When I tried to run , I got the below screen:    The captcha is missing!  I tried in other browsers too.    Update:  From @Oded's comments, I changed to HTTP and it was working perfectly.  Out of curiosity I would like to know why the captcha isn't working when accessing Data Explorer over HTTPS. Proving $(P(AB))^2+(P(AB^c))^2+(P(A^cB))^2+(P(A^cB^c))^2\ge\frac{1}{4}$ for any events $A$ and $B$ Show that for any two events $A$ and $B$, $$(P(AB))^2+(P(AB^c))^2+(P(A^cB))^2+(P(A^cB^c))^2\ge\frac{1}{4}$$   Using the Inclusion-Exclusion formula, De Morgan's law and $P(AB^c)=P(A)-P(AB)$, I was able to expand the $\rm{l.h.s.}$ as $$(x+y)(x+y-4z)+2z(2z+1)+x(x-2)+y(y-2)+1\quad...\quad(*),$$ where $x=P(A),y=P(B)$ and $z=P(AB)$.  However I'm not being able to group the terms involved in the $\rm{l.h.s.}$ to form a 'suitable' expression and probably the way I have grouped them is not correct.  The expression $(*)$ is subject to the condition   $\max\{0,x+y-1\}\le z\le \min\{x,y\}$ and obviously $0\le x,y,z\le 1$, but I am unable to use this.      Is there an elegant/slick way to show the inequality? It probably can be shown without a lot of work and it's possible I'm missing something trivial here. Here at the Genealogy and Family History Stack Exchange we see many questions about relationship mapping which is also known as:   relationship calculation kinship terminology etc   If you are just trying to understand how to calculate "cousins and removes" then  provides an excellent guide.  However, if you are seeking an English term for a more complicated relationship between two members of an extended family, rather than asking a question like:          site users are encouraged to first review the answers below.    Note: This Q&amp;A arises from  and is set to  to help make it canonical. Remove empty elements from an array in Javascript How do I remove empty elements from an array in JavaScript?   Is there a straightforward way, or do I need to loop through it and remove them manually? How can I show a supremum of a set is also its limit point? It just seems like I can't wrap my head around this. For a set $A$ how can I show that the supremum of that set $\sup(A)$ is also a limit point of said set? Can I just set $$a=\sup(A)$$ $$b\le a \: \:  \forall \: b \in A$$ and then show that there's a point different from $a$ in any $\epsilon$-neighbourhood for any $\epsilon&gt;0$? Or should I use a completely different approach? I have a hard time finding a how-to.  I would like to set up a simple ESD mat and wrist wrap, but I am not sure about is the the 1 MOhm resistor should be, if at all.  Several options:   1 MOhm from ground outlet to the mat, wire from the common earthing on the mat to the wrist wrap. 1 MOhm from ground outlet to the mat, wire from another contact on the mat to the wrist wrap. 1 MOhm from ground outlet to the mat, 1 MOhm from the same mat contact to the wrist wrap. wire from ground outlet to the mat, 1 MOhm from the same mat contact to the wrist wrap. wire from ground outlet to the mat, wire from another mat contact to the wrist wrap.   I am not sure why the 1 MOhm wire is needed since the mat itself already has that kind of resistance (as long as the wrist wrap does not connect to ground directly, but through the mat).  Which solutions above are ok? Which is best? Should I do something else instead? How to call C++ function from C? How can I make a bash script executable by double clicking just like .exe files in Windows? I tried , but there are two consequences:   the terminal twinkles, disappears, and nothing is done. you must specify to run in terminal in order to work.   I have a script that installs tomcat on an offline PC, including all dependencies of tomcat in the script. I need to make the script work on double clicking like windows since most who use the script will not be familiar with Ubuntu.  Forget the above explanation. I want to make a script that can be run by double-clicking on it, without using the terminal. Anybody knows how? $\lim_{n \rightarrow\infty} ~ x_{n+1} - x_n= c , c > 0$ . Then, is $\{x_n/n\}$ convergent? When I change the stroke width of a rectangle the width and and height change also. How can I  prevent this from happening? What does "all the more" mean and how is it used? Here is a sample context:     As you might expect, the framework also provides some convenience   classes to make the task of writing custom validators all the more agreeable.   (Emphasis added.) What are Harry, Ginny, Ron and Hermione's professions? Simulating Key Press C# My Minecraft app now only has the option of "play" and there is no "play on realms" to be found!  I downloaded - and paid for - the multiplayer app, but there doesn't seem to be a way to get to any realms on that app either. I feel uncomfortable saying sentences like the following:    "I've a car" instead of "I have a car" "They've a great time" instead of "They have a great time" "He's a pen" instead of "He has a pen" etc   I ask this because I read this sort of thing in a book.  Are they correct? And what is the rule? Can you use such forms in a formal setting? Automatic field should take the required entry The characteristic and minimal polynomial of a companion matrix of a monic polynomial $f \in \mathbb F\left[x\right]$ in $1$ variable $x$ over a field $\mathbb F$ plays an important role in understanding the structure of finite dimensional $\mathbb F[x]$-modules.  It is an important fact that the characteristic polynomial and the minimal polynomial of $C(f)$ are both equal to $f$. This can be seen quite easily by induction on the degree of $f$.  Does anyone know a different proof of this fact? I would love to see a graph theoretic proof or a non inductive algebraic proof, but I would be happy with anything that makes it seem like more than a coincidence! Is it possible to load some package from a .sty file only if some option is selected?  (Naive approach \DeclareOption{something}{\RequirePackage{something}} doesn't work.) How should different sized RAM be added to iMac? I have a mid-2011 iMac which came with 4GB RAM (2x2GB). I just bought 8 GB (2x4GB) and added it to get a total of 12GB. The current configuration is as follows:    Given that the RAM is a different size, do I have the optimal locations for the different RAM chips? At the moment what I did was to simply add the 2 new sticks of RAM in the two empty slots.