Relationship between Binomial distribution and the Beta distribution I have been investigating the details of the Beta distribution and the Binomial distribution and have 2 questions to ask, but first a slight preamble to explain the background to my questions. In the following, I am assuming that the normal parameters for the Beta and Binomial distribution ($\alpha, \beta, n, k$) are positive integers greater than zero, and $x$ is the target probability that we are looking for.  So, in my investigation, I started with the equation for the Beta distribution in the form  $Beta(x|\alpha,\beta) = \frac {\Gamma (\alpha +\beta )}{\Gamma (\alpha) \Gamma(\beta )}x^{\alpha-1}(1-x)^{\beta-1} $ and the Binomial distribution in the form $Bin(x|n,k) = {\binom {n}{k}}x^{k}(1-x)^{n-k}$. Then using the fact that $\Gamma(x+1) \equiv x!$, I re-worked the equations for the Beta distribution and the Binomial distribution and came to the following set of four relationships between the two. They are all essentially the same, just expressed in different formats, i.e. going from Beta to Binomial, or Binomial to Beta, and interchanging their normal parameters. To get to the four relationships, the following intermediate relations were derived from the equivalence of the Beta and Binomial distributions:  $\alpha = k + 1; \beta = n - k + 1; n = \alpha + \beta  - 2; k = \alpha - 1$  The four relationships are:   $Beta(x|\alpha,\beta) = Bin(x|\alpha+\beta-2, \alpha-1)*(\beta+\alpha-1)$ $Beta(x|\alpha, \beta) = Bin(x|n,k) * (n+1)$ where $ n = \alpha + \beta  - 2$ and $ k = \alpha - 1$ $Bin(x|n,k) = \frac{Beta(x|k+1, n-k+1)}{n+1}$ $Bin(x|n,k) = \frac{Beta(x|\alpha, \beta)}{\alpha+\beta-1}$ where $\alpha = k + 1$ and $\beta = n - k + 1$   I tested these numerically and they agree, at least for the number set I used. As the above description is quite condensed, and I am happy to elaborate on how I reached these conclusions if required.  So  my questions are: 1. Is my analysis correct, and the conclusion that the Binomial and Beta distributions are closely linked as shown above? 2. If it is correct, can anyone point me to any references for more detail on these relationships?
Relationship between Binomial and Beta distributions I'm more of a programmer than a statistician, so I hope this question isn't too naive.  It happens in sampling program executions at random times. If I take N=10 random-time samples of the program's state, I could see function Foo being executed on, for example, I=3 of those samples. I'm interested in what that tells me about the actual fraction of time F that Foo is in execution.  I understand that I is binomially distributed with mean F*N. I also know that, given I and N, F follows a beta distribution. In fact I've verified by program the relationship between those two distributions, which is  cdfBeta(I, N-I+1, F) + cdfBinomial(N, F, I-1) = 1   The problem is I don't have an intuitive feel for the relationship. I can't "picture" why it works.  EDIT: All the answers were challenging, especially @whuber's, which I still need to grok, but bringing in order statistics was very helpful. Nevertheless I've realized I should have asked a more basic question: Given I and N, what is the distribution for F? Everyone has pointed out that it's Beta, which I knew. I finally figured out from Wikipedia () that it appears to be Beta(I+1, N-I+1). After exploring it with a program, it appears to be the right answer. So, I would like to know if I'm wrong. And, I'm still confused about the relationship between the two cdfs shown above, why they sum to 1, and if they even have anything to do with what I really wanted to know.
Currently, when someone suggests an edit, they are unable to keep track of it without watching reputation levels or constantly refreshing the page. What I would like is a notification, that shows up like the notification where someone has edited your question, that shows whether a suggested edit has been approved or rejected after the last vote is cast. This way, the community can be notified when their edit has been rejected/approved, while focusing their efforts on another question/answer. Why prefer the 18-55mm and 55-250mm lenses vs 18-200mm? Identifying Unused Stored Procedures This next year, I am helping an effort to clean several SQL Server environments.    We have about 10,000 stored procedures and estimate that only about 1000 of them are used on a regular basis, and another 200 or so are used on a rare occasion, meaning we have a lot of work to do.    Since we have multiple departments and teams that can access these databases and procedures, we are not always the ones calling the procedures, meaning that we must determine what procedures are being called.  On top of that, we want to determine this over a few months, not in a few days (which eliminates some possibilities).  One approach to this is to use the SQL Server Profiler and track what procedures are being called and compare them to the list of what procedures we have, while marking whether the procedures are used or not.  From then, we could move the procedures to a different schema in case a department comes screaming.  Is using the Profiler the most effective approach here?  And/Or have any of you done something similar and found another way/better way to do this? Prove that the difference between two rational numbers is rational This is a terribly simple question I'm sure, but I can't find a work-around in my proof. I must prove that the difference between two rational numbers is thus rational. Here is my attempt:     Let $a$ and $b$ be rational numbers. Therefore, \begin{align}  a=\frac{\lambda}{\beta},\:b=\frac{\xi}{\zeta},\:\ni\:\lambda,\beta,\xi,\zeta\in\mathbb{Z},\tag{1} \end{align} which gives us \begin{align}  a-b=\frac{\lambda}{\beta}-\frac{\xi}{\zeta}=\frac{\lambda\zeta-\xi\beta}{\beta\xi}.\tag{2} \end{align}   So I have shown that $a$ and $b$ are rational numbers which, by definition, can be represented by the quotient of two integers. But now how do I tackle the problem of the difference? By definition the difference between two integers is an integer. Does that require that this difference is thus rational?  Thank you for your time, Clicking on a spoiler tag to reveal the text shouldn't navigate you to another page This is an issue I noticed when reading . On multiple occasions I ended up inadvertently clicking a link that I could not see when I was just trying to reveal the spoiler text.         Can links be deactivated in spoilers until the text is revealed?  (Edit: this issue seems particularly to apply to those using touch screen devices with the "full site" theme) Parameterize an SQL IN clause Is this mold? Black? White? Guys I need to find out what this is- it's also on my wooden bed racks I was wondering what might be making my questions less attractive to users?  I get very low views most of the time. What should be done to increase a question's popularity? How do you search through flags in the manpages? Whenever I search the manpages, I'm almost always trying to find out what's going on with a specific flag. For example, when searching the man pages for  I was searching the for -f flag. I'm not sure how to say, quickly get to the flag which is here:  OPTIONS    General options        -f FORMAT, -r FORMAT, --from=FORMAT, --read=FORMAT   and filter out stuff like:  iconv -t utf-8 input.txt | pandoc | iconv -f utf-8              or  pandoc -f markdown -t latex hello.txt   I've tried searching for /-f, and I've looked at  and tried (^|\s)-f($|\s\) but it was invalid. So I'm wondering how do you quickly search less for "only the flag -f"?  Would love your thoughts or some other tool that you use to quickly go through documentation Suppose I have BaseClass with public methods A and B, and I create DerivedClass through inheritance.  e.g.  public DerivedClass : BaseClass {}   Now I want to develop a method C in DerivedClass that uses A and B.  Is there a way I can override methods A and B to be private in DerivedClass so that only method C is exposed to someone who wants to use my DerivedClass? Heisenberg famously derived his uncertainty principle by considering the disturbance that a measurement would have on a small enough system.  Of course in the mathematical formalism of Quantum Mechanics the relationship is derived from more basic principles.  How does String Theory account for it? Heuristically, and mathematically? PHP mail function doesn't complete sending of e-mail Particle effect showing only on one side of mirrored object I'm trying to create a fur particle effect on sonic the hedgehog, and I'm using weight paints to keep it in the areas I want.   Problem: I can only get the hair effect to show up on certain random areas on model, and only on one side.  I'm using a mirror effect to keep model symmetrical, but hair only shows on one side of the mirror.  I'm new used Blender, so I don't know what all the settings for the particle effects do.  Whats going on and how do I fix it? In the context of topological spaces, I see the following major countability properties:  A space is:   "separable" iff it has a countable dense subset "second countable" iff if has a countable basis "first countable" iff the neighbourhood system of every point has a countable local basis.   (Definitions taken from Counterexamples in Topology by Steen and Seebach, 2nd ed. 1978 -- there may be differences in wording from other sources.)  The question I have is: "separable" into what, exactly? By which I mean to say: what is the thinking behind calling such a condition "separable"?  It appears not to be related to the concept of "separation axioms", which do immediately and obviously invoke an intuitive notation of separation, neither does it seem to have anything to do with "separated sets".  (Anyone using the spelling "seperable" or "seperated" will be immediately downvoted. :-) ) What areas of physics depend on the sum $1 + 2 + 3 + 4 + 5 + 6+ 7+\ldots= -1/12$? This youtube video from Numberphile,   shows how the value is derived. In the video, one interviewee claims that "this result is used in many areas of physics". In the video, only string theory is mentioned.  Which areas of physics use or depend on the sum $$1 + 2 + 3 + 4 + 5 + 6+ 7+\ldots= -1/12?$$ string array with garbage character at end I have a char array buffer that I am using to store characters that the user will input one by one.  My code below works but has a few glitches that I can't figure out:   when I execute a printf to see what's in Buffer, it does fill up but I get garbage characters at the end it won't stop at 8 characters despite being declared as char Buffer[8];   Can somebody please explain to me what is going on and perhaps how I could fix this? Thanks.  char Buffer[8]; //holds the byte stream int i=0;  if (/* user input event has occurred */)  {         Buffer[i] = charInput;         i++;          // Display a response to input         printf("Buffer is %s!\n", Buffer);  }   Output:   tagBuffer is 1┬┬w! tagBuffer is 12┬w! tagBuffer is 123w! tagBuffer is 1234! tagBuffer is 12345! tagBuffer is 123456=! tagBuffer is 1234567! tagBuffer is 12345678!  tagBuffer is 123456789! System of linear equations having a real solution has also a rational solution. Entropy and the principle of least action Is there any link between the law of maximum entropy and the principle of least action. Is it possible to derive one from the other ? I'm a bit confused about the concept of ground, and perhaps voltage as well, particularly when trying to analyze a circuit. When I learned about Ohm's law in grade school, I learned how to apply the law to calculate current, voltage, and resistance of simple circuits.   For instance, if we were given the following circuit:    We would be could be asked to calculate the current passing through the circuit. At the time, I'd simply compute (based on the rules given) 1.5V/1Ohms=1.5A.  Later on, however, I learned that the reason the voltage of the resistor would be 1.5V is because voltage is really the difference in potential between two points, and that the difference of the voltage across the battery would be the same as that of the resistor (correct me if I'm mistaken), or 1.5V. I got confused, however, after the introduction of the concept of ground.   The first time I tried to do the current calculation for a circuit similar to the previous circuit on a simulator, the program complained about not having a ground and "floating voltage sources". After a bit of searching, I learned that circuits need ground as a reference point or for safety reasons. It was mentioned in one explanation that one can pick any node for ground, although it's customary to design circuits so there is a "easy place" to pick ground.   Thus for this circuit    I picked ground at the bottom, but would it be okay to pick ground between the 7 ohm and 2 ohm resistor - or any other place? And what would be the difference when analyzing the circuit?  I've read that there are 3 typical ground symbols with different meanings - chassis ground, earth ground, and signal ground. A lot of circuits I've seen used in exercises either use earth ground or signal ground. What purpose is there in using earth ground? What is the signal ground connected to?  Another question: since the ground is at unknown potential, wouldn't there be current flowing to or from ground to the circuit? From what I've read we treat the ground as 0V, but wouldn't there be some sort of effect because of a difference in potential of the circuit and ground? Would the effect be different depending on what ground was used?  Finally: In nodal analysis, one customarily picks a ground at the negative terminal of the battery. However, when there are multiple voltage sources, some of them are "floating". What meaning does the voltage of a floating voltage source have? Error -504 on appliation installation Yes. I've read all other questions regarding this problem. Like this one:  It's the same problem as this one but with more background information and an other app:   The details: My device is an Ulefone Be Touch 2 running Android 5.0.  My connection is perfect. To be sure I tried different networks, among them are 3G, 4G and wifi on ADSL, VDSL, cable and fiber. I've also used Tor and 3 different VPN providers in 3 countries on 2 continents. Let's say I ruled the network out.   A single app does not install: OpenKeychain.  Currently I have 126 apps installed. It does not install via desktop browser > play store > install button.    I have enough free space (40GB). The app is 7MB.   Re-adding Google account and clearing caches did not work.    Even a complete device wipe and re-install didn't work.     Downloaded the apk and tried manual installation, same error.   I also own a Nexus 7 (2013 wifi) and the app installs fine on that device.