How to display double values properly in c++?
How do I print a double value with full precision using cout?
Fibonacci combinatorial identity: $F_{2n} = {n \choose 0} F_0 + {n\choose 1} F_1 + ... {n\choose n} F_n$ Can someone explain how to prove the following identity involving Fibonacci sequence  $F_n$  $F_{2n} = {n \choose 0} F_0 + {n\choose 1} F_1 + ... {n\choose n} F_n$ ? Say you take a dice and you roll it twice so that you have a pair (X,Y) where X represents the first roll, Y represents the second.   When you have a distribution like max(X,Y), what type of distribution is that? It's not uniform, obviously, because the outcomes of max(X,Y) are clearly more favorable towards the higher numbers. I'm being asked to "describe" the distribution. I know that we can look at each distinct outcome, like:  max(1,1) = 1  max(1,2) = 2  max(1,3) = 3  max(1,4) = 4  max(1,5) = 5  max(1,6) = 6  max(2,1) = 2  max(2,2) = 2  max(2,3) = 3  max(2,4) = 4  max(2,5) = 5  max(2,6) = 6  max(3,1) = 3  max(3,2) = 3  max(3,3) = 3  max(3,4) = 4  max(3,5) = 5  max(3,6) = 6  max(4,1) = 4  max(4,2) = 4  max(4,3) = 4  max(4,4) = 4  max(4,5) = 5  max(4,6) = 6  max(5,1) = 5  max(5,2) = 5  max(5,3) = 5  max(5,4) = 5  max(5,5) = 5  max(5,6) = 6  max(6,1) = 6  max(6,2) = 6  max(6,3) = 6  max(6,4) = 6  max(6,5) = 6  max(6,6) = 6  But is that the most effective way to look at a distribution like this? Is there a faster and more concise way to describe this distribution other than saying the probability of each outcome?  i.e.,  P(max = 1) = 1/36  P(max = 2) = 3/36 = 1/12  P(max = 3) = 5/36  P(max = 4) = 7/36  P(max = 5) = 9/36 = 1/4  P(max = 6) = 11/36  So, obviously, I understand how to LOOK at a discrete distribution and see what the probabilities are, but not until I look at each possible outcome. But what does it mean to "describe" a distribution? This isn't uniform or any sort of discrete distribution we're taught. Can someone provide clarity? I made a plot but it's not at all what I want. There are two things I'm having trouble with: 1.) The bars should be at the bottom, but there's a space between them and the x-axis. 2.) E1/E4, E2/E5 and E3/E6 should have the same color.  Here's a picture of how it is and how I want it below:   And here's my code:  \documentclass{standalone} \usepackage{pgfplots,pgfplotstable} \begin{document} \begin{tikzpicture} \begin{axis}[ symbolic x coords={E1, E4 (dyn.), E2, E5 (dyn.), E3, E6 (dyn.)}, xtick=data, width=\linewidth, height=6cm ] \addplot[ybar,fill=blue, bar width = 1cm] coordinates {     (E1,   8)     (E2,  15)     (E3,   50)     (E4 (dyn.), 4)     (E5 (dyn.), 17)     (E6 (dyn.), 40)  }; \end{axis} \end{tikzpicture} \end{document}   I hope you can help me. Thanks. Do I have to care about bad boxes? When I compile my PDF from LaTeX source, do I have to care about all the messages (currently 28) about 'bad boxes'?  The PDF seems fine to me. Start Ubuntu Unity With Dash Open conjunction of disjunctions and disjunction of conjunctions $(p∨q) ∧ (p∨r) ∧ (q∨r) \iff (p∧q) ∨ (p∧r) ∨ (q∧r)$  Without using set theory, how can I prove this using logical equivalences? I've tried to approach this problem with De Morgan and I've also tried the property of distributivity but I keep reaching a dead end.   How should I approach this problem?  Thank  you. Taking back turns, and forgotten privileges What's the fastest connection speed / shortest elapsed time using Migration Assistant? "By" vs "Per". Which one should I use on expressions like "P&L/Geography" or "VaR/Asset"? In the finance field, it is quite common to express some measures in relation to some grouping criteria. Some groupings can be temporal (year, month, etc.) and others like greography, asset class (e.g. a financial product), strategy (used in trading), and others.   Which one is the most appropriate preposition to apply in this kind of expression ?    "VaR per asset" or "VaR by asset" "P&amp;L per strategy" or "P&amp;L by strategy".   Is there any well defined rule behind the most appropriate use ? Split a file by line and have control over resulting files extension If $n$ vectors are linearly independent, is their span $\mathbb{R}^n$? Have $n$ vectors in $\mathbb{R}^n$.  If the $n$ vectors are linearly independent, can we conclude that their span is $\mathbb{R}^n$? How to recover files from dead windows using dos? I use both GNOME and KDE SC. In GNOME, I use Docky but I do not use it with KDE. So, I would like Docky to start up only when I login into GNOME not KDE. In KDE, there is an option to make a program autostart only in KDE. But I cannot find any such option in GNOME.    How can I do this?    Thanks in advance. Mac App Store: Downloads just don't start (Mojave) I want to download apps in the Mac App Store and the apps won't start to download. As you can see in the pictures below, the spinner starts spinning but nothing gets downloaded, the download simply doesn't start. I am logged in with my account. I tried logging out of the App Store and logging in again, restarting the Mac etc, nothing helped.  This is a relatively fresh installation of Mojave 10.14. Before with High Sierra 10.13 everything worked fine. It's a Late 2013 MacBook Pro 13". Installations are allowed from App Store and identified developers.   Some screenshots from Console output below. I am trying to put a server together to install Ubuntu. I bought an Intel S1200v3RPS motherboard and plugged an Intel Xeon E3 processor into it. I purchased a 1U chassis from Supermicro with a power supply.  The power supply does not have a 5-pin cable to the motherboard's Auxiliary Power Supply socket. A bit of research did not give any definite answer on the need for the 5-pin connection. The motherboard does not power on and the CPU fan is off. I tested the power supply on a desktop Intel motherboard and it works fine.   I would like to buy a new power supply but could not find one which expressly stated that it comes with the 5-pin aux power supply cable. Any recommendations on a power supply unit for this motherboard would be helpful. Any comments on the problem would be greatly appreciated. Show that if $n$, $a$, $b$, and $c$ are positive integers with $\gcd(a, b) = 1$ and $ab = c^n$, then there are positive integers $d$, and $e$ such that $a = d^n$ and $b = e^n$.   I know that (by Bezout) $\gcd\left(a,b\right) = 1$ implies $ax + by = 1$ for some integers $x$ and $y$, and also that $\gcd\left(a^n,b^n\right) = 1$, but this does not help me. Proving $\frac{1}{\sqrt{x}}\ge \frac{2}{x+1}$ for $x> 0$ Prove: $$\frac{1}{\sqrt{x}}\ge \frac{2}{x+1}, \quad\forall x&gt;0$$  Yeah, pretty much it. I've tried all manner of rearranging and just can't seem to get it. Thanks. QGIS 2.0 - can't select items in Print Composer Install a .NET windows service without InstallUtil.exe Practical hyperparameter optimization: Random vs. grid search