Drawing curved midarrows in diagram
How to draw this P-V graph?
How do I make multiple spawn points in minecraft vanilla? Prediction interval for binomial random variable What is the formula (approximate or exact) for a prediction interval for a Binomial random variable?  Assume $Y \sim \mathsf{Binom}(n, p)$, and we observe $y$ (drawn from $Y$). The $n$ is known.  Our goal is to obtain a 95% prediction interval for a new draw from $Y$.   The point estimate is $n\hat{p}$, where $\hat{p}=\frac{y}{n}$. A confidence interval for $\hat{p}$ is straightforward, but I cannot find a formula for a prediction interval for $Y$. If we knew $p$ (rather than $\hat{p}$), then a 95% prediction interval just involves finding the quantiles of a binomial. Is there something obvious I'm overlooking? OpenLayers low resolution and/or shifts in QGIS print composer? Why am I not burned by a strong wind? Extending from two classes Is something similar to Robin's theorem known for possible exceptions to Lagarias' inequality? Robin's theorem says that if $$\sigma(n)&lt;e^\gamma n\log\log n$$ holds for all $n&gt;5040$, where $\sigma(n)$ is the sum of divisors of $n$, then the Riemann hypothesis is true, but if there are any counterexamples, then they are colossally abundant numbers, and there are infinitely many counterexamples.  Lagarias' theorem says $$\sigma(n)\leq H_n+e^{H_n}\log(H_n)$$ holding for all natural numbers $n$ is equivalent to the Riemann hypothesis, where $H_n$ is the $n$th harmonic number, the sum of the reciprocals of the first $n$ positive integers. It looks like Lagarias' inequality is sharper than Robin's, so any counterexamples to Robin's inequality must also be counterexamples to Lagarias'. Is known to be impossible for integers that aren't colossally abundant numbers or of some similar more general type to be exceptions to Lagarias' inequality? How to change 'et al' to italic font in citation call-outs, with 'agsm' bibliography style Finding information on German soldiers from World War I and World War II? In a perfect world, the so-important advice "do not reinvent the wheel" would be extensively followed. In practice, everyday millions of wheels are reinvented simply because someone didn't know an existing solution for a specific problem. The best way to find a library, today, is googling it, but it's not always obvious which keywords to use. The lack of a tool for finding solutions is very counterproductive for the dev world as a whole. Stack Overflow is not very receptive for recommendation threads. So what about a new SE for it? What does "mostly" modify? "My friends are mostly non-smokers."  Is "mostly" a focusing adverb that modifies "non-smokers"? Are there any tools that I can use to defrag my Windows partition from Linux? How to prevent Google from finding my admin index page? food safety regarding re freezing twice cooked chicken Let $P$ be a prime. Show that $\exists$ $x \in \mathbb{N}$ such that $f(x) = p$ then $\exists$ $y \in \mathbb{N}$ such that $g(y) = p$ What is given?  $$\text{Let P be a prime}$$ $$\text{Let} \space f(x)= 3x+1$$ $$\text{Let} \space g(x)= 6x+1$$  Show that: If there exists $x \in \mathbb{N}$ such that $f(x) = P $ , then there exists a $y \in \mathbb{N}$ , such that $g(y) = P$  What have I tried?   Given the question says 'show that', that leads me to believe I need to do a proof of sorts. I'm guessing it might be a proof by either strong induction, or just induction (or maybe just a direct proof?).  I don't even know how to set this up If $f(x) =$ $3x + 1 = $ prime...for any integer $x$, then there exists an integer $y$ such that $g(y) = 6y + 1 = $ prime?   Any hints or starting points would be greatly appreciated.   $$\text{Thanks in advance!}$$ How to delete specific lines from a txt file I have a Bash script that writes data into a log file, then from that I take out stuff I want using the sed command. That gives me a file data.txt. What I want to know is how I go about deleting specific line inside that data.txt file.  For example:  123 456 789   I want to remove the 2nd line, containing 456 so that I only have  123  789   I tried with sed '2d' data.txt but it doesn't work.  I use this command to create data.txt from log.log:  sed -nE '1s/.{1}(.{2}).*/\1/p;' log.log &gt;&gt; data.txt Limit of the sequence $a_n=\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}$ p-value approach - clarify teacher's solution? So I have a question with my teacher's solution below and I'm confused about one of the steps in their solution. I would appreciate some clarification.  Two brands of coffee were compared. Two independent random samples of 50 people each were asked to taste either Brand A or Brand B coffee, and indicate whether they liked it or not. Eighty four percent of the people who tasted Brand A liked it; the analogous sample proportion for Brand B was ninety percent.   (A) At α = 0.01, is there a significant difference in the proportions of individuals who like the two coffees? Use the p-value approach.   So I circled in blue a step where I'm confused about in the teacher's solution. I don't understand why P(Z&lt;-0.89) = 0.2061.   On the z-score table -0.89 is equal to 0.1867 so shouldn't it be that value instead? where did 0.2061 come from? Is it possible that the teacher made an error or am I not understanding something? Orthogonal matrix and orthonormal columns Prove: if $a,b\in G$ commute with probability $>5/8$, then $G$ is abelian Suppose that $G$ is a finite group. If $P( ab=ba ) &gt;5/8$, prove $G$ is abelian. If I put my phone on my notebook's left corner, it shuts down. Can anybody give me reliable explanation as to why this is happening??