In the option table for  there is a column called 'Price' and a column called 'Bid' and a column called 'Ask'.  If I wish to buy a call option, I need to pay a premium. Which column is the premium I would pay? Is it the 'Price'? But isn't the 'Price' supposed to be mid-way between the 'Bid' and 'Ask'? Why wouldn't it be?
In my online brokerage account, I want to buy a particular stock and I see the following:  Bid:  13.20 x200           Ask:  13.27 x1,000   the current stock price is 13.22.   Can someone explain what the bid and ask prices mean relative to the current price?  If I buy 1000 shares, why would I pay more?
I compiled the following file (through MiKTeX which I have just updated all files), and I do not get the junk.pdf file inserted.  \documentclass{report} \usepackage{pdfpages} \begin{document} qwe \includepdf[pages=-]{./junk.pdf} \end{document}   The .log and junk.pdf file are there: ;   OK, yes it works with pdflatex, but I have to use latex->dvips->ps2pdf because of other constraints insertion of EPS figures). Beside, I do not know how to use the suggested lastpage option.  Using \includepdf[nup=2x2,frame,pages={1-13}]{./junk.pdf} I do correctly get 5 pages which should have included junk.pdf, but I get blank pages with a black dot in the center. If $f$ is bounded and $\lim_{x \to c} g(x) = 0$, then $\lim_{x \to c}g(x)f(x) = 0$ How do I view all available HDD's/partitions? I am trying to prove  $$\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$$  This problem is a classic, but I seem to be missing one step or the understanding of two steps which I will outline below.   $$I_n := \int\cos^n x \ dx = \int\cos^{n-1} x \cos x \ dx \tag{1}$$  First question: why rewrite the original instead of immediately integrating by parts of $\int \cos^n x \ dx$?  Integrate by parts with $$u = \cos^{n-1} x, dv = \cos x \ dx \implies du = (n-1)\cos^{n-2} x \cdot -\sin x, v = \sin x$$  which leads to  $$I_n = \sin x \ \cos^{n-1} x +\int\sin^2 x (n-1) \ \cos^{n-2} x \ dx \tag{2}$$  Since $(n-1)$ is a constant, we can throw it out front of the integral:  $$I_n = \sin x \ \cos^{n-1} x +(n-1)\int\sin^2 x \ \cos^{n-2} x \ dx\tag{3}$$  I can transform the integral a bit because $\sin^2 x + cos^2 x = 1 \implies \sin^2 x = 1-\cos^2 x$  $$I_n = \sin x \ \cos^{n-1} x + (n-1)\int(1-\cos^2 x) \ \cos^{n-2} x \ dx \tag{4}$$  According to Wikipedia as noted , this simplifies to:  $$I_n = \sin x \ \cos^{n-1} x + (n-1) \int \cos^{n-2} x \ dx - (n-1)\int(\cos^n x) \ dx \tag{5}$$  Question 2: How did they simplify the integral of $\int(1-\cos^2 x) \ dx$ to $\int(\cos^n x) \ dx$?  Assuming knowledge of equation 5, I see how to rewrite it as  $$I_n = \sin x \ \cos^{n-1} x + (n-1) I_{n-2} x  - (n-1) I_{n} \tag{6}$$  and solve for $I_n$. I had tried exploiting the fact that  $$\cos^2 x = \frac{1}{2} \cos(2x) + \frac{1}{2} $$  and trying to deal with $\int 1 \ dx - \int \frac{1}{2} \cos (2x) + \frac{1}{2} \ dx$  which left me with $\frac{x}{2} - \frac{1}{4} \sin(2x)$ after integrating those pieces. Putting it all together I have:  $$I_n = \sin x \ \cos^{n-1} x + (n-1) I_{n-2} x \left(-(n-1) (\frac{x}{2} - \frac{1}{4} \sin 2x) \right) \tag{7}$$  but I'm unsure how to write the last few terms as an expression of $I_{something}$ to get it to match the usual reduction formula of   $$\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$$ $x^y = y^x$ for integers $x$ and $y$ Inheriting constructors How should references in non-Latin script language be treated in English paper? Let's say I want to cite a Russian text in my own paper which is written in English. Should I:   reference it as is, in the original Cyrillic alphabet, although the readers won't be able even to pronounce it? transliterate the authors' names, while keeping the rest as is? transliterate the names and translate the title of the work, journal name, etc. into English? Accessing inherited variable from templated parent class Consider the following code:  template&lt;class T&gt; class Foo { public:   Foo() { a = 1; }  protected:   int a; };  template&lt;class T&gt; class Bar : public Foo&lt;T&gt; { public:   Bar() { b = 4; };    int Perna(int u);  protected:   int b;  };  template&lt;class T&gt; int Bar&lt;T&gt;::Perna(int u)  {    int c = Foo&lt;T&gt;::a * 4; // This works   return (a + b) * u;    // This doesn't }   g++ 3.4.6, 4.3.2 and 4.1.2 give error   test.cpp: In member function `int Bar&lt;T&gt;::Perna(int)': test.cpp:25: error: `a' was not declared in this scope   g++ 2.96 and MSVC 6, 7, 7.1, 8 and 9 accept it, as do (at least) older Intel and SGI c++ compilers.   Do the new Gnu C++ compiler obey the standard or not? If they do, what is the rationale behind that inheriting class is not able to see a protected inherited member variable?  Also, if there's  int A() { return a; }    in Foo, I get error  test.cpp:25: error: there are no arguments to A that depend on a template parameter, so a declaration of A must be available test.cpp:25: error: (if you use -fpermissiveâ, G++ will accept your code, but allowing the use of an undeclared name is deprecated)   when I try to use it in a member function of Bar. I find that curious as well: Bar inherits Foo, so I think it's obvious that A() in scope of Bar is Foo::A(). Test Class Apex Page Parameters Interpretation of R's lm() output The help pages in R assume I know what those numbers mean, but I don't. I'm trying to really intuitively understand every number here. I will just post the output and comment on what I found out. There might (will) be mistakes, as I'll just write what I assume. Mainly I'd like to know what the t-value in the coefficients mean, and why they print the residual standard error.   Call: lm(formula = iris$Sepal.Width ~ iris$Petal.Width)  Residuals:      Min       1Q   Median       3Q      Max  -1.09907 -0.23626 -0.01064  0.23345  1.17532    This is a 5-point-summary of the residuals (their mean is always 0, right?). The numbers can be used (I'm guessing here) to quickly see if there are any big outliers. Also you can already see it here if the residuals are far from normally distributed (they should be normally distributed).  Coefficients:                  Estimate Std. Error t value Pr(&gt;|t|)     (Intercept)       3.30843    0.06210  53.278  &lt; 2e-16 *** iris$Petal.Width -0.20936    0.04374  -4.786 4.07e-06 *** --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1    Estimates $\hat{\beta_i}$, computed by least squares regression. Also, the standard error is $\sigma_{\beta_i}$. I'd like to know how this is calculated.  I have no idea where the t-value and the corresponding p-value come from. I know $\hat{\beta}$ should be normal distributed, but how is the t-value calculated?  Residual standard error: 0.407 on 148 degrees of freedom   $\sqrt{ \frac{1}{n-p} \epsilon^T\epsilon }$, I guess. But why do we calculate that, and what does it tell us?  Multiple R-squared: 0.134,  Adjusted R-squared: 0.1282    $ R^2 = \frac{s_\hat{y}^2}{s_y^2} $, which is $ \frac{\sum_{i=1}^n (\hat{y_i}-\bar{y})^2}{\sum_{i=1}^n (y_i-\bar{y})^2} $. The ratio is close to 1 if the points lie on a straight line, and 0 if they are random.  What is the adjusted R-squared?  F-statistic: 22.91 on 1 and 148 DF,  p-value: 4.073e-06    F and p for the whole model, not only for single $\beta_i$s as previous. The F value is $ \frac{s^2_{\hat{y}}}{\sum\epsilon_i} $. The bigger it grows, the more unlikely it is that the $\beta$'s do not have any effect at all. Converting ISO 8601-compliant String to java.util.Date Is there any UNIX variant on which a child process dies with its parent? Does installing many programs really slow down Windows? Will running a large query on a secondary database in an availability group affect transaction performance in the primary database? I need to provide real-time, or almost real-time, data for SSRS and Tableau reporting. I don't want the production OLTP system to be negatively impacted by long running queries. Will running a large query on a secondary database in an availability group affect transaction performance in the primary database? Expected value in an urn problem In an urn there are $n$ red balls and $m$ blue balls. I extract them without replacement. Let $X$=time of first blue. What is $E(X)$?  I found PMF of $X$ and it is, if $k &gt; n+1$  $$P(X=k) =\frac{n(n-1) \dots (n-k+1)m}{(m+n)(m+n-1) \dots (n+m-k+1)}$$  How could I evaluate $E(X)$?  Edit: I'm looking for an explicit form of $E(X)$ du shows completely different folder size for * and folder name I'm trying to move some data over and to start with I'd like to make sure that the size is ok. I run into a problem however, if I check the folder with   du -sh  myfolder/ 780M    myfolder/   but   du -sh * .. 71M     myfolder ..   I thought it might be something to do with hidden files but this particular folder does not have any inside.  Am I missing something? Photoelectric effect in a a frequency range I know how to calculate the kinetic energy of the emitted electrons   $$KE=hf-\phi$$  And the number of the emitted electrons, but this is valid for a certain frequency  $$N=\frac{IA}{hf}=\frac{P}{hf}$$  What happens if I have a frequency range? for example between λ1 and λ2?  BTW: It's not homework, I'm an engineering student,  not a physics student did + present tense (or ) Past tense of the verb What did you say?   What you said.         How did you and mum guess I wanted Agnes?   how you and Mum guessed I wanted Agnes   What is the meaning of above two sentences? When should I use "did + present tense" and the past tense of a verb? Relating the union of the power set of a set to the set itself Activating 'missing' standard GRASS utilities in QGIS Processing toolbox