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How to construct an example of a probability distribution for which $\mathbb{E}\left(\frac{1}{X}\right)=\frac{1}{\mathbb{E}(X)}$ holds, assuming $\mathbb{P}(X\ne0)=1$?    The inequality which follows from Jensen's inequality for a positive-valued RV $X$ is like $\mathbb{E}\left(\frac{1}{X}\right)\ge\frac{1}{\mathbb{E}(X)}$ (the reverse inequality if $X&lt;0$). This is because the mapping $x\mapsto\frac{1}{x}$ is convex for $x&gt;0$ and concave for $x&lt;0$. Following the equality condition in Jensen's inequality, I guess the distribution has to be degenerate for the required equality to hold. A trivial case where the equality holds is of course if $X=1$ a.e. Here is an example I found in a problem book: Consider a discrete random variable $X$ such that $\mathbb{P}(X=-1)=\frac{1}{9}, \mathbb{P}(X=\frac{1}{2})=\mathbb{P}(X=2)=\frac{4}{9}$. It is then easily verified that $\mathbb{E}\left(\frac{1}{X}\right)=\frac{1}{\mathbb{E}(X)}=1$.   This example shows that $X$ need not be positive (or negative) a.e. for the equality in the title to hold. The distribution here is not degenerate either.  How do I construct an example, possibly like the one I found in the book? Is there any motivation? find an approximate solution, up to the order of epsilon The question is to find an approximate solution, up to the order of epsilon of following problem.  $$y'' + y+\epsilon y^3 = 0$$ $$y(0) = a$$ $$y'(0) = 0$$  I tried to solve the given problem using perturbation theory.  $$y(t) = y_0(t) + \epsilon y_1(t) + \epsilon^2 y_2(t) + \cdots$$ $$t = w(\epsilon)t = (1 + \epsilon w_1 + \epsilon^2 w_2 + \cdots )t$$  However, i failed to find an appropriate approximate solution... help me!! Any positive linear functional $\phi$ on $\ell^\infty$ is a bounded linear operator and has $\|\phi \| = \phi((1,1,...)) $ How to change Status Bar text color in iOS How does one parse XML files? Booting from USB and not installing I created a live usb to boot Kubuntu, which has worked well enough. However, when the installation gets to the point of telling me to restart the computer (and it restarts) the installation window pops up again, and the computer is still signed in as a live user. None of the installation or settings appear to have saved.  Any ideas? This happens pretty much every time I try to bake normal maps. It's quite a broad error message and I still don't know exactly what the problems I'm having are, only that if I keep clicking things for half an hour eventually it sometimes just works.  My understanding of the things needed:   A "Generated" blender Image that the normals are written to (In UV/Image Editor, click Image->New)   Tell blender that this is the bake output, below (this is the really confusing bit which blender seems to try to guess and fails)  A UV Map for the low poly model (you only need the geometry of the high poly and it doesn't need a UV map) Under Bake, in the rendering tab:   Bake Mode = Normals Normal Space = Tangent Selected to Active = ✓ (for the next bit)  Select the high rez object first, then the low poly object With the two objects selected, click Bake (you may have to move the mouse to see the error pops up)   Now, from what I can tell the only  (with regard to the setup and apart from the internals of baking) is how blender finds the output image:   Cycles  Have a material using an ImageTexture node which points to your normal map output. Then  says the node must be selected, but  doesn't mention selecting. Blender Render  Have a texture attached to the material that uses the image.  is pretty vague on assigning the output image. Sometimes I've found hitting tab to go to edit mode and selecting the normal map image in the UV editor to help, or at least it seemed to.   I would like to know the exact details of what blender goes looking for when you hit Bake, rather than have to guess by following tutorials. For example, it seems to look for the high and low poly objects in your selection and based off the order you select them. How does it find the normal map/image to write to for both cycles and blender render? Does the Texture need to Influence Normals or does it just need to reference the Image? etc... In general, what is meant by saying that the fraction $x$ of the variance in an analysis like PCA is explained by the first principal component? Can someone explain this intuitively but also give a precise mathematical definition of what "variance explained" means in terms of principal component analysis (PCA)?  For simple linear regression, the r-squared of best fit line is always described as the proportion of the variance explained, but I am not sure what to make of that either. Is proportion of variance here just the extend of deviation of points from the best fit line? Which derivation of drift velocity is correct? In the derivation of drift velocity I have seen two variations and want to know which one's correct.   $s=ut+\frac{at^2}{2}$ Assume that the drift velocity of any electron in any conductor is : $$\vec{v_d}=\frac{\vec{l}}{t}$$ Due to the electric field the acceleration of electrons in any conductor is: $$a=\frac{-e\vec{E}}{m}$$ Now the distance travelled by an electron after a long time (initial thermal velocity = 0) $$l=\frac{at^2}{2}\implies \frac{-e\vec{E}t^2}{2m}$$ the time between the collisions is $\tau$ $\therefore$ $$l=\frac{-e\vec{E}\tau^2}{2m}$$ thus the velocity is $$\vec{v_d}=\frac{-e\vec{E}\tau}{2m}$$ In another proof I saw the author using $v=u+at \implies v_d=\frac{-e\vec{E}\tau}{m}$   My question is which of the the two equations of motion can be used in the proof? Can they be used at all. I want to be able to calculate the confidence interval from the estimated coefficient and respective standard errors.  I have a linear regression model which can be summarized (in R):  summary(fit1)  Call: lm(formula = bwt ~ height + weight + parity, data = data)  Residuals:     Min      1Q  Median      3Q     Max  -66.913 -10.624   0.991  10.979  55.621   Coefficients:             Estimate Std. Error t value Pr(&gt;|t|)     (Intercept) 31.19217   13.56879   2.299   0.0217 *   height       1.24964    0.23083   5.414 7.48e-08 *** weight       0.06781    0.02823   2.402   0.0164 *   parity1     -1.83309    1.19838  -1.530   0.1264     --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  Residual standard error: 17.9 on 1170 degrees of freedom Multiple R-squared:  0.04898,   Adjusted R-squared:  0.04654  F-statistic: 20.08 on 3 and 1170 DF,  p-value: 1.071e-12   With this model I can calculate the respective confidence intervals:  &gt; confint(fit1)                   2.5 %     97.5 % (Intercept)  4.57029503 57.8140351 height       0.79676227  1.7025207 weight       0.01243198  0.1231932 parity1     -4.18429933  0.5181151   I would expect the intervals of the predictor height to be given by  $$ 1.24964 \pm (1.96*0.23083) = [0.7972132,1.702067] $$  where 1.24964 is the estimated value for the coefficient and 0.23083 is the standard error for this coefficient. The numbers are close but not quite the same.  What am I doing wrong? What are attributes in .NET? How does the Comma Operator work How does the comma operator work in C++?  For instance, if I do:  a = b, c;     Does a end up equaling b or c?   (Yes, I know this is easy to test - just documenting on here for someone to find the answer quickly.)  Update:  This question has exposed a nuance when using the comma operator.  Just to document this:  a = b, c;    // a is set to the value of b!  a = (b, c);  // a is set to the value of c!   This question was actually inspired by a typo in code.  What was intended to be  a = b; c = d;   Turned into  a = b,    //  &lt;-  Note comma typo! c = d; Best way to center a  on a page vertically and horizontally? What packages do you use and recommend for creating graphics in your LaTeX documents?     As this is a community wiki post, please add your package to the accepted answer (or add a comment, and someone with >100 rep will add it to the CW answer), and include a brief description of what differentiates it from others and how it can be used (GUI drawing tool which generates code, type in raw text, or generates image for inclusion in document).   We'll eventually sort these answers under headings. How to run a LaunchAgent that runs a script which causes failures because of System Integrity Protection After upgrading to Mojave, my rsync based backup script run via a launch agent in ~/Library/LaunchAgents, could no longer read some directories in ~/Library. I'm learning Neural Networks and I grasped the algebra behind them. I'm now interested in understanding how normalization and centering of the input data affect them. In my personal learning project (Regression with NN) I've transformed my input variables to a range between 0 and 1 using the following function:  normalize &lt;- function(x) {return((x - min(x))/ (max(x) - min(x)))}   The NN model fits well and has an acceptable out-of-sample prediction error.   However, I read in other  that scaling the inputs to have mean 0 and a variance of 1 is advised for NN. I don't fully understand:   how this transformation works better for NN against the min-max normalization between 0 and 1. how can I assess which transformation to apply in my data? The adjective ones I have heard recently are    forward straightforward forthcoming frank   I was wondering if each of them has positive, negative or neutral meaning?  What are other similar terms that are used for positive, neutral and negative intention? How to get my own IP address and save it to a variable in a shell script? How can I get my own IP address and save it to a variable in a shell script? Behaviour of printf when printing a %d without supplying variable name I've just encountered a weird problem, I'm trying to printf an integer variable, but I forgot to specify the variable name, i.e.  printf("%d");   instead of  printf("%d", integerName);   Surprisingly the program compiles, there is output and it is not random. In fact, it happens to be the very integer I wanted to print in the first place, which happens to be m-1.  The errorneous printf statement will consistently output m-1 for as long as the program keeps running... In other words, it's behaving exactly as if the statement reads  printf("%d", m-1);   Anybody knows the reason behind this behaviour? I'm using g++ without any command line options.  #include &lt;iostream&gt; #define maxN 100 #define ON 1 #define OFF 0  using namespace std;  void clearArray(int* array, int n); int fillArray(int* array, int m, int n);  int main() {     int n = -1, i, m;     int array[maxN];     int found;      scanf("%d", &amp;n);      while(n!=0)     {         found=0;         m = 1;         while(found!=1)         {             if(m != 2 &amp;&amp; m != 3 &amp;&amp; m != 4 &amp;&amp; m != 6 &amp;&amp; m != 12)             {                 clearArray(array, n);                 if(fillArray(array, m, n) == 0)                 {                     found = 1;                 }             }             m++;         }          printf("%d\n");          scanf("%d", &amp;n);     }      return 0; }  void clearArray(int* array, int n) {     for(int i = 1; i &lt;= n; i++)         array[i] = ON; }  int fillArray(int* array, int m, int n) {     int i = 1, j, offCounter = 0, incrementCounter;      while(offCounter != n)     {         if(*(array+i)==ON)          {             *(array+i) = OFF;             offCounter++;                }         else          {             j = 0;             while((*array+i+j)==OFF)             {                 j++;             }             *(array+i+j) = OFF;             offCounter++;                    }         if(*(array+13) == OFF &amp;&amp; offCounter != n) return 1;         if(offCounter ==n) break;          incrementCounter = 0;                while(incrementCounter != m)         {             i++;             if(i &gt; n) i = 1;             if(*(array+i) == ON) incrementCounter++;          }            }      return 0; } How can I identify hardware components in a laptop? I just reinstalled my Windows and now I don't have all the hardware components recognized. I have an  laptop and  website seems to have all the necessary drivers for my model. The problem is that I don't know exactly which components I have. For example, in my laptop video card case the site has available drivers for ATI and Intel chips.   How do I know what are the right drivers for my machine?