Given two independent random variables $X_1\in NB(p,r_1)$ and $X_2\in NB(p,r_2)$, find the distribution of random variable $X_1+X_2$   I came up with an answer that is obviously wrong since it's different from what's on the textbook ( $X_1+X_2 \in NB(p,r+s)$), how do I go about showing this?
Suppose $X, Y$ are independent random variables with $X\sim NB(r,p)$ and $Y\sim NB(s,p)$. Then $$X + Y \sim NB(r+s,p)$$  How do I go about proving this? I'm not sure where to begin, I'd be glad for any hint.
is there a way to find suitable ratio for matches to controls (using "ratio" in Matchit package in R or other programs) I am using matchit package in R to find a control set for my treated units and I am testing the model with different ratio to get suitable control set. Is there a way to find suitable ratio and I will use that to find my control set. How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$ If $f(x)$ is a continuous function on $(-\infty, +\infty)$ and $\int_{-\infty}^{+\infty} f(x) \, dx$ exists. How can I prove that $$\int_{-\infty}^{+\infty} f(x) \, dx = \int_{-\infty}^{+\infty} f\left( x - \frac{1}{x} \right) \, dx\text{ ?}$$ How can we prove that every odd perfect square is congruent to $1$ modulo $8$? I'd like to generate unique random numbers between 0 and 1000 that never repeat (i.e. 6 doesn't show up twice), but that doesn't resort to something like an O(N) search of previous values to do it. Is this possible? Can generated OpenVPN keys be used on multiple clients? We are experimenting with running an OpenVPN server for our business. One question I can't seem to find the answer to is this:  When we generate keys for one of our users for them to use at home, can their use the same keys on their home laptop as well as their home desktop? Or do we need to generate separate keys for each user's client machine? I had an application for returning closest matches to certain values in a large  cluster of values( as in my earlier )   and I chose a VBA solution. While in the course of using the said application, I observed that the results for the value of 0.5 were not correct. I had been using the VBA Round function which I found to be returning 0 for 0.5 rounded to integer whereas the worksheet round function returns 1. Strangely, the VBA round function returns 2 for 1.5. I had to substitute the worksheet function in place of the VBA one.  Am I missing something? Any way to tell apart a CPU defect from a mainboard defect? I have a fairly modern desktop computer, an AMD Athlon II X2 based silent PC, that does not work.  When turned on, it will start physically (all fans are rotating, disks start up...) but not give a signal on any of the graphics ports (DVI, VGA, and HDMI, I tried all three). Also, the reset button does not seem to have any effect.  I have stripped the mainboard bare of all SATA connections, extension cards, and the one 2GB RAM chip to eliminate them as the problem source, but to no avail, so I think it's fair to assume it's either the mainboard or the processor that are at fault.  However, I have neither a replacement mainboard, nor a replacement processor handy to identify which one is broken. The cause of the defect is unknown, so for all I know, it could be both. Therefore, I'm reluctant to buy replacement hardware blindly before knowing more.   Is there any way to further diagnose (or at least get some indication) which component is broken without buying replacement hardware? Why am I asked for a credit card number to download a free app? I  want to download free app in my iPod touch device and i have the following doubts.  Kindly resolve me doubts.   when i try to download it asks my credit card security number. since i am trying to download free app , why do i need to provide my credit card security code? Will i be debited for downloading the free app from app store?   I cant able to find any information regarding this on apple.  Please provide me some references about this from apple and please resolve my doubts.  Thanks a lot. What is the best base to use? Make it possible to retract comment flags I misread the content of a comment, and flagged it. Then I realized that the comment wasn't what I had initially thought. Would it be possible to have the ability to retract comment flags, just like we have the ability to retract flags on questions and answers?  I'm aware of this &#32;. But the answer there is just that it's not possible. At the time it was not possible to retract any flags.   This is a separate  which requests that it may become possible to retract comment flags.  If I've realized a flag was a mistake, I think it's unnecessary to keep it in the queue of content the moderators have to look at. Also, it would add an unnecessary "rejected" to the flaggers statistics. Sure, people should think before they flag. But everyone makes mistakes sometimes. Evaluating $\int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x$ I was generalizing the following integral: $$\int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x \hspace{40pt} n\geq 0$$ We can start by noting that $\sin x =\dfrac{e^{ix}-e^{-ix}}{2i}$ and thus $\displaystyle \sin^{2n+1}x=\frac{(-1)^n}{2^{2n}}\sum_{r=0}^n (-1)^r \binom{2n+1}{r}\sin(2r+1)x$. This implies $$\begin{aligned}\displaystyle \int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x &amp;=\frac{(-1)^n}{2^{2n}}\sum_{r=0}^n (-1)^r \binom{2n+1}{r}\int_{0}^{\infty}\frac{\sin(2r+1)x}{x}\mathrm{d}x \\ &amp;=\frac{(-1)^n\pi}{2^{2n+1}}\sum_{r=0}^n(-1)^r\binom{2n+1}{r} \\ &amp;=\frac{(-1)^n\pi}{2^{2n+1}}\sum_{r=0}^n\left((-1)^r\binom{2n}{r}-(-1)^{r-1}\binom{2n}{r-1}\right)\end{aligned}$$ Where I have used the well known  $\displaystyle \int_{0}^{\infty} \frac{\sin(2r+1)x}{x}\mathrm{d}x=\int_{0}^{\infty}\frac{\sin x}{x}\mathrm{d}x=\frac{\pi}{2}$ and the property of binomial coefficients that $\displaystyle \binom{n}{r}+\binom{n}{r-1}=\binom{n+1}{r}$ Since the above sum telescopes, we have $$\displaystyle \int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x=\frac{(-1)^n\pi}{2^{2n+1}}(-1)^n\binom{2n}{n}=\frac{\pi}{2^{2n+1}}\binom{2n}{n} ~\forall ~ n\in \mathbb{Z^{+}}$$ I would like to know other methods for evaluating this integral. As title,  Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?  Say if $K/F$ is a finite extension when $K$ is a finite-dimensional vector space of $F$. Clearly, this implies that $K$ is finitely generated (as an algebra) over $F$, since a basis is a generating set. So every finite extension is finitely generated.  So indeed they all are, is my logic correct? Storage for Eldritch Horror Is there an optimal strategy for storing Eldritch Horror cards and tokens so that game play is facilitated.   I am not looking for answers that including "zip bags" and other things to keep the game element safer while is storage. Although, a strategy that works well for both cases will clearly be superior to one that only works for game play.  I am not limiting this to commercial solutions. I am quiet happy to build your custom made solution if you release the specification/plans of course!  Although, it it requires a 3D printer, more power to you. show that a distance function is continuous Let $X$ be a metric space with metric $d$. Define $d: X \times X \to \mathbb{R}$, show that $d$ is continuous.  I would like to show that the function is continuous the topology way (since it is a course on topology).   Let $(a,b)$ be a basic open set in $\mathbb{R}$, then $d^{-1}(a,b) = \{(x,y): a&lt;d(x,y)&lt;b\}$. Define $A= \bigcup_{x \in X} B_{b} (x)$, where $B_b (x) = \{y \in X| d(x,y)&lt;b$}. Clearly $A$ is open. Now define $C = \{(x,y):d(x,y)&gt;a \}.$ Essentially I hope that $A \cap C$ is open. But I am stuck in showing that $C$ is open.  Any help please? :) Why is the Cantor function not absolutely continuous? Is there an easy way to see that the Cantor function is not absolutely continuous that directly uses the definition of absolutely continuous? Mean Value Theorem: $\frac{2}{\pi}<\frac{\sin x}{x}<1$ I need to show that $\dfrac{2}{\pi}&lt;\dfrac{\sin(x)}{x}&lt;1$ for $0&lt;x&lt;\dfrac{\pi}{2}$.  I know I need to use the mean value theorem, would I just say that since $f$ is continuous in the interval (call it I).  We know $\exists c\in I$ such that $f'(c) = \dfrac{f\left(\dfrac{\pi}{2}\right)-f(0)}{\dfrac{\pi}{2}}$. I don't know where this is going though.  Also, would I need to show $\lim_{x \to 0}\dfrac{\sin(x)}{x} = 1$? Is there a way to do this without L'H Rule? Slave duplicate counter Is it possible to create a counter that is a "slave" to another (master) counter, but also an exact duplicate?  The \@addtoreset{&lt;slave&gt;}{&lt;master&gt;} resets &lt;slave&gt; whenever &lt;master&gt; is incremented (similar functionality is offered by ), but I don't know about anything incrementing one counter (&lt;slave&gt;) automatically whenever another counter (&lt;master&gt;) is incremented.  The idea is to have a macro (say) \dupcntr{&lt;slave&gt;}{&lt;master&gt;} as an analogue to \@addtoreset. The application for this might be where one is interested in renaming a counter used/provided by a package into something more meaningful for the user (say, &lt;slave&gt;), without having to reference &lt;master&gt;. Or, if the original counter contains @ that needs to be escaped every time it's used, it is sometimes more convenient to have a "more user friendly" counter.  I know one way around is to use a macro that returns the value of &lt;master&gt;, but is it possible to perform the same manipulation with a macro than one could with the counter? Manipulation here refers to the printing of the counter (via \the&lt;slave&gt;) and perhaps it's representation (like \arabic, \alph, ...). Is 2h 45m enough for a layover in Newark Liberty International transferring from an international to a domestic flight (Terminal B → Terminal C)? Changed response to Recommendation questions now that there's a site for them? How can blackbody radition be explained by quantization? I don't understand why quantization makes a peak on the blackbody radiation curve (so there is no ) and the relationship between that peak and quantization concept.  When the blackbody is heated, it starts to glow. All atoms start to vibrate. Total heating energy must be divided and equally shared by all atoms of blackbody. Right?  But what happens next?  Do some atoms get much more energy to radiate at higher frequency and waste the energy? What happens at the atomic level, so quantization can explain it? What does classical physics claim, so it can not explain the radiation curve?