Given $N$ Bernoulli random variables $X_i$ that take on value 1 with probability $p = 0.4$ and $0$ otherwise, what is the probability that $Y = 1$, where $$\begin{align} Y &amp;=\lim_{N\to\infty} Y_N\\ &amp;= \lim_{N \to \infty} X_1 \oplus X_2 \oplus \cdots \oplus  X_N \end{align}$$ The answer is apparently 0.5, but I am having trouble figuring out why. I think the approach must be to derive some equation for $Y_N$ and then take the limit, and this expression, would be recursive in nature as $Y_N$ can be obtained from $Y_{N+1}$. Is this the idea, or is there something really simple that I'm missing?
Suppose $X$ is binomial $B(n,p)$. How can I find the probability that $X$ is even ?  I know $$P(X = k ) = \frac{ n!}{(n-k)!k!} p^k(1-p)^{n-k} $$  where $X=1,....,n$. Are they just asking to find $P(X = 2m )$ for some $m &gt; 0$ ?
How widely-accepted is "What do you got?" to Americans? Watching , I noticed that Melanie Griffith twice asked    I recognise this as an American construction which sounds strange to me — Brits invariably say either "What have you got?" or "What do you have?". But I'd be interested to know if it's considered "normal" by most/all Americans, or if it's regionally or otherwise restricted. Forcing a SimpleXML Object to a string, regardless of context QGIS - export a project with dependencies What's the best way to invite a user to answer a question on Stack Overflow? On several occasions, I was reading a question thinking "I know a guy that knows the answer to this" and that guy (or gal) also happens to be a registered user. How can I bring this to the attention of this user? When conducting multiple regression, when should you center your predictor variables & when should you standardize them? In some literature, I have read that a regression with multiple explanatory variables, if in different units, needed to be standardized.  (Standardizing consists in subtracting the mean and dividing by the standard deviation.)  In which other cases do I need to standardize my data?  Are there cases in which I should only center my data (i.e., without dividing by standard deviation)? Help finding this set Lets define the following:  Let A be a set. A is innumerable if and only if there exists a bijective function from A to $\mathbb{N}$  Proof that there exists an innumerable set $B \subseteq \mathcal P \left({\mathbb{N}}\right)$ such that it has the following properties:  1) $X\cap Y =\emptyset$ for all $X,Y \in B$ such that $X \neq Y$  2) $\mathbb{N}= \bigcup B$  3) All elements of B are innumerable.  Any help is appreciated. I don´t know how to find this set ... If $U$ is unitary operator then spectrum $\sigma(U)$ is contained inside the unit circle In a Hilbert space, let $U$ be a continuous operator which it unitary. Prove $\sigma (U)\subseteq \Bbb{S}^1$.   It is important for me to know how I am doing, and I didn't come by a clear explanation so it would be appreciated to have your correcting and guiding.  Attempt: $U^{*}U=UU^{*}=I$ and therefore $U^{-1}=U$. I have $$ \| U x \|^2 = \langle U x, U x \rangle = \langle U^* U x, x \rangle = \langle x, x \rangle = \|x\|^2 $$ and the same goes for $U^*$ and therefore $\| U \| = \| U^{-1} \| = 1$.   Looking at $U-\lambda$ I can take $$ -\lambda U(U^{-1}-{1\over \lambda}I) =U-\lambda I. $$ Clearly, for $\lambda \ne 0$, $-\lambda U$ is defined and is a linear operator. $(U^{-1}-{1\over \lambda}I)$ is invertible if $|\lambda|\ge 1$ and therefore $\sigma (A^{-1})=\sigma (A)\subseteq \{|z|\le1\}$. But from the RHS, $U-\lambda I$ is invertible if $|\lambda|\le 1$ and therefore $\sigma(A)\subseteq \Bbb{S}^1$. I have noticed my claims tend to have holes in them many times which I cannot spot. If it is substantially true, how can I make sure it is completely formal? I believe it is what the Germans call "". English itself has no such equivalent word. (Although it has been adopted as a .)  Does an idiom exist that describes it? Why is part of my model not being rendered? Why is the selected object displaying in the 3D view, but not rendering?  In object mode, it shows up:   However, it does not show up in render mode:   Why is this so and what causes it to hide in render mode? How to select JSF components using jQuery? Does Blender support dual-GPU rendering if the two GPUs aren't the same? I have a GTX 1060 6GB in my current build, and I also have a GTX 960 2GB in storage. I was wondering if I got , and got both of my GPUs in one build (not SLI or CrossFire, just in the same system) would I be able to use both GPUs for rendering simultaneously? I don't care if this means having them both render separate tiles in one instance of Blender, or rendering separate frames in two instances. I'm using Cycles for everything I do. What are the rules for splitting words at the end of a line? What are the rules in English language to split words at the end of a line?   Where exactly must the hyphen split the word? In Helgason, "Differential Geometry, Lie Groups and Symmetric Spaces"  we have the formula for the exterior derivative  $(p+1)d\omega(X_{1}, \ldots X_{p+1})=\Sigma_{i=1}^{p+1} (-1)^{i+1} X_{i} \cdot \omega(X_{1}, \ldots , \hat{X_{i}}, \ldots X_{p+1})+\Sigma_{i&lt;j} (-1)^{i+j}\omega([X_{i}, X_{j}], X_{1}, \ldots , \hat{X_{i}}, \ldots \hat{X_{j}}, \ldots , X_{p+1})$  but in other sources on-line they have the same formula without the factor of $p+1$ on the left-hand side.  Are there two different definitions of the exterior derivative, or is there an error in Helgason? The div containing tags for a question doesn't reflow, causing layout issues for questions with long/many tags This appears to be a css bug which (I think) has recently been introduced. It's most visible when viewing questions on Stack Exchange sites using mobile devices (the full site with responsiveness, not the mobile site) but can easily be reproduced by resizing a browser window on a question where the combined width of the tags is large. A couple of example questions show this behaviour:  On Meta Stack Overflow:   On Meta Stack Exchange:     Using the Meta Stack Overflow question to demonstrate (as the total width of the tags is larger):  At 1100px, the tags just about fit into their containing div.     At 800px, the tags appear on top of related questions:     At 450px, the tags are now wider than the view (the last tag is completely off the screen), causing scroll bars to appear:      This doesn't happen to lists of questions - the tags flow onto multiple lines:     Reproduced on: { Firefox 81, Chrome 84, Edge 84 } on Windows 10, and Chrome 85 on Android 11. Is it possible to make armour that can survive /kill? /kill is said to be infinity damage but I have a question can you really survive it in Java edition. However, it is possible that you can pretty much survive this if you are on Minecraft Java Edition in 1.8 or lower and you can just have resistance 5 and you will survive this but can you make armour that helps you survive in 1.16 and in java-edition. Dominoes and induction, or how does induction work? I've never really understood why math induction is supposed to work.  You have these 3 steps:   Prove true for base case (n=0 or 1 or whatever) Assume true for n=k.  Call this the induction hypothesis. Prove true for n=k+1, somewhere using the induction hypothesis in your proof.   In my experience the proof is usually algebraic, and you just manipulate the problem until you get the induction hypothesis to appear.  If you can do that and it works out, then you say the proof holds.  Here's one I just worked out,  Show $\displaystyle\lim_{x\to\infty} \frac{(\ln x)^n}{x} = 0$  So you go:   Use L'Hospital's rule. $\displaystyle\lim_{x\to\infty} \frac{\ln x}{x} = 0$. Since that's $\displaystyle\lim_{x\to\infty} \frac{1}{x} = 0$. Assume true for $n=k$. $\displaystyle\lim_{x\to\infty} \frac{(\ln x)^k}{x} = 0$. Prove true for $n=k+1$.  You get $\displaystyle\lim_{x\to\infty} \frac{(\ln x)^{k+1}}{x} = 0.$   Use L'Hospital again: $\displaystyle\lim_{x\to\infty} \frac{(k+1)(\ln x)^{k}}{x} = 0$.  Then you see the induction hypothesis appear, and you can say this is equal to $0$.  What I'm not comfortable with is this idea that you can just assume something to be true ($n=k$), then based on that assumption, form a proof for $n=k+1$ case.  I don't see how you can use something you've assumed to be true to prove something else to be true. Do I need to tame my twitchy F5 finger? I just got this error message for the first time (I was getting this as the response to any request on the SE network for about the last five minutes, until now):     We're sorry...      There are an unusual number of requests coming from this IP address.   To protect our users, we can't process any more requests from this IP   address right now. We'll restore access as quickly as possible, so try   again soon. If you believe you have reached this page in error,   contact us.   As far as I know, I was not spamming SE or participating in DOS attack, and I find it unlikely that any of my office coworkers who share my IP address were doing so either.  Granted, I was pressing F5 a lot. But I've always pressed F5 a lot on SE, and have never gotten this message before. So my question is: Is it possible to get this error message simply by refreshing too much or too often? Or is that message reserved for a truly massive number of requests that are unlikely to be caused by one person refreshing a lot? (In the former case, I can't explain why I've never gotten it before; in the latter case, I can't explain why I just got it now). I have searched through the internet and in AskUbuntu, but I could not find a working solution for my little problem.  The RTL8723ae WLAN is behaving strangely on Ubuntu 14.04 and drops the WLAN connection constantly. I am using the notebook MSI GE40 and have already seen on the web that there are other people having the same problem but there was no permanent solution. The problem did not exists on the original Windows 8 on the device and the router is a TP-Link, which I can control (channel change and so on, I use WPA2-PSK WLAN security). When should I use double or single quotes in JavaScript? console.log(&quot;double&quot;); vs. console.log('single'); I see more and more JavaScript libraries out there using single quotes when handling strings. What are the reasons to use one over the other? I thought they're pretty much interchangeable. automatic size adjustment for nested parentheses As I prefer outer parentheses to grow larger in nested expressions, I happen to insert \bigl and \bigr and its larger cousins a lot. I have always wondered whether there is a way to do this automagically.  Inserting \left and \right prophylacticly before all opening and closing parentheses and hoping that it sorts things out doesn't do the trick in all cases as seen below.  The left column is without any size changing commands. The middle column as a \left \right pair at every parentheses/brace/bracket and the rightmost column shows my preferred version.    As seen in the middle column of the second and third line the \left and \right pair sometimes makes the outer parentheses larger, on the cost of an extra horizontal space between \Pr and (. The extra space is fine in case of really big opening parentheses but here I'd prefer the regular spacing.  \documentclass{scrartcl} \usepackage{amsmath} \usepackage{amsfonts} \begin{document} \begin{align*} &amp;1-(1-F(x))^n&amp;&amp;1-\left(1-F(x)\right)^n&amp;&amp;1-\bigl(1-F(x)\bigr)^n\\ &amp;\Pr(X_{(1)}\le x)&amp;&amp;\Pr\left(X_{(1)}\le x\right)&amp;&amp;\Pr\bigl(X_{(1)}\le x\bigr)\\ &amp;\mathbb{E}[\min\{X_1,X_2\}]&amp;&amp;\mathbb{E}\left[\min\left\{X_1,X_2\right\}\right]&amp;&amp;\mathbb{E}\bigl[\min\left\{X_1,X_2\right\}\bigr]\\ &amp;\left(\pi-\arccos(\frac{y}{r})\right)&amp;&amp;(\pi-\arccos\left(\frac{y}{r}\right))&amp;&amp;\left(\pi-\arccos \left(\frac{y}{r}\right)\right) \end{align*} \end{document}   As mathematical expressions can be arbitrarly complex there is probably no general way to do this. But I’m not asking for a solution of the general case. A partial solution that works in the simple cases shown above would be a big help. The simple rule could be, that parentheses never shrink below the size of an inner pair. Of course you can immediately think of extensions like a \mid in a conditional expectation that grows with the size of the expectation’s brackets.  What is your preference regarding the size of nested parentheses? Is there a better way to archieve your preferred style despite inserting \bigl \bigr manually?    EDIT:  A colleague pointed me to section 8.3 of Herbert Voß’  document where size problems with parentheses are solved by playing with two TeX parameters within a group around the expression in question. This led naturally to appendix G of the TeXbook where the mechanics of \delimitershortfall and \delimiterfactor are explained.  \delimitershortfall specifies the maximum space not covered by a delimiter (default 5pt) and \delimiterfactor is the ratio for variable delimiters, times 1000 (default 901).  I used them to implement the never shrink below a subformular size idea from above by setting the shortfall to 0pt and the ratio to 1.0.    While it works nicely in line one and three, in line two and four the parentheses now grow too much, and still there is the extra horizontal space introduced by \left.