Let $X$ and $Y$ are random variables and they are both independent from the $\sigma$-algebra $\mathscr F$. Given a Borel measurable function $f(x,y)$, is $f(X,Y)$ also independent from $\mathscr F$?  I guess it is right, but actually not quite sure. Here is what I do:  Let $A$ be a Borel set, then $B = \{ f(X,Y) \in A \}$ is also a Borel set. I consider the case where $B = B_1 \times B_2$ is a rectangular area. Then given $E \in \mathscr{F}$, $P(\{f(X,Y)\in A \}\cap E) = P((\{ X \in B_1\} \cap E) \cap (\{Y \in B_2\} \cap E))$. It seems I cannot proceed any more. And I even get skeptical whether it is right.  Any help is appreciated!
Let $X_1,...,X_n$ be mutually independent RVs. Suppose $X_i \perp \mathcal F $ for $1\ \leq \forall i \leq n$.    How can I show that:  $$\sigma (X_1,...,X_n) \perp \mathcal F$$ ?  What I have tried:  $\sigma (X_1...X_n) = \sigma(\cup_{i=1} ^n \sigma(X_i))$ holds. I think I need some Dynkin's lemma-ish argument but can't see how.
The strongest player in the game is a close ally of mine and we each founded and follow of one of the strongest two religions in the game, with little cross-over between religions within our territories.  I've just spotted that he has a Great Prophet approaching my territory. According to the Civopedia they are powerful at spreading religion, and I'd rather not have a huge hit to the followers of my religion in my cities.  How can I prevent another players Great Prophet from damaging the religion in my cities, preferably without causing a major diplomatic incident. How to generate all permutations of a list? How do you generate all the permutations of a list in Python, independently of the type of elements in that list?  For example:  permutations([]) []  permutations([1]) [1]  permutations([1, 2]) [1, 2] [2, 1]  permutations([1, 2, 3]) [1, 2, 3] [1, 3, 2] [2, 1, 3] [2, 3, 1] [3, 1, 2] [3, 2, 1] Banner ads should not be shown on the right-hand ad slot Can I use Mage Hand to fly by Polymorphing myself and picking myself up with the hand? Could I first cast Mage Hand, then Polymorph into a Tiny creature less than 10 pounds then use the Mage Hand to pick myself up and fly at 30 Ft per turn?  Polymorph says:     The new form can be any beast ... The target's game statistics,   including mental ability scores, are replaced by the statistics of the   chosen beast. It retains its alignment and personality ... The   creature is limited in the actions it can perform by the nature of its   new form and it can't speak, cast spells, or take any other action   that requires hands or speech.   This keeps me from casting spells, which keeps me from just casting Mage Hand to fly as long as I want to over and over. But for that initial duration of the Mage Hand cantrip, could I use the Mage Hand to fly as a Tiny polymorphed creature until the spell runs out? When should I use exposure compensation? When should I use exposure compensation, rather than ISO, shutter speed or aperture? "NG NS" on ticket after airport security A question from Kunen's book: chapter VII (H9), about diamond principle Why do we need to use int main and not void main in C++? Visual Studio - How to change the return value of a method in the debugger? When I'm debugging, I often have to deal with methods that does not use an intermediate variable to store the return value :     private int myMethod_1()    {       return 12;    }     private int myMethod_2()    {       return someCall( someValue );    }   Well, in order to recreate a bug, I often have to change values on the fly. Here, I could want to see what happens when myMethode_1 return zero. Same thing for myMethod_2.  Is there a way to do that without modifying the code ? What I would like to do, it's to place a breakpoint on the return line (or on the closing bracket) and type-in a new return value.  Note : I'm using Visual Studio 2005 and Visual Studio 2008.  Thank ! How to use the Alt + Right click in edit mode in Linux What is the physical significance of dot & cross product of vectors? Why is division not defined for vectors? I get the physical significance of vector addition &amp; subtraction. But I don't understand what do dot &amp; cross products mean?   More specifically,    Why is it that dot product of vectors $\vec{A}$ and $\vec{B}$ is defined as $AB\cos\theta$? Why is it that cross product of vectors $\vec{A}$ and $\vec{B}$ is defined as $AB\sin\theta$, times a unit vector determined from the right-hand rule?   To me, both these formulae seem to be arbitrarily defined (although, I know that it definitely wouldn't be the case).   If the cross product could be defined arbitrarily, why can't we define division of vectors? What's wrong with that? Why can't vectors be divided? If $0$ is the zero-object $ \Longrightarrow F(0) $ is the zero object when $F$ additive The Fourier transform of a "comb function" is a comb function? Let $f(x) = \sum_{n=-\infty}^{\infty} \delta(x - n)$, where $\delta$ is the Dirac delta function. This function $f$ (a "comb function") is important in signal processing because evenly sampling a function $g$ can be viewed as multiplying $g$ pointwise with $f$. This idea is the first step towards understanding how to approximate the Fourier transform of $g$, given evenly spaced samples of $g$. The next step is to note that $\hat{gf} = \hat{g}*\hat{f}$, where $\hat{f}$ is the Fourier transform of $f$ and $*$ denotes convolution.  Is it true that $\hat f$ is also a comb function? If that statement isn't quite right, what is the correct statement?  I would like to see: 1) an informal, intuitive, nonrigorous but easy derivation of the Fourier transform of $f$ . 2) A rigorous version of the same calculation.  (How would you even define $f$ rigorously, is it a distribution?)  (I'd also be interested in recommendations of math textbooks that cover this topic, including the Nyquist sampling theorem, even if it's only an exercise or series of exercises in an analysis textbook.) How does a  work, and what differentiates it from other linear classifiers, such as the , , or ? *  (* I'm thinking in terms of the underlying motivations for the algorithm, optimisation strategies, generalisation capabilities, and run-time complexity) Change the default editor when sudo visudo When doing sudo visudo, the default editor is nano. I want to change it to vi or vim.     I already made vim the default editor, and use it as a substitute for gedit to open php, .txt, .c and .h files, by placing a vim.desktop file in ~/.local/share/applications/ and by correctly editing either  ~/.local/share/applications/mimeapps.list or /etc/gnome/defaults.list. However apparently this does not apply to nano.    Any clue ? SSH and home directory permissions How to move Ubuntu to an SSD My current situation is:   One hard disk Dual boot Ubuntu 11.04 and Windows 7. Partitions:  100MB Windows System thingy 144GB Main Windows 160GB Ubuntu 4GB Swap 12GB System Restore stuff    Now I want to install an 80GB SSD and move Ubuntu to it. AFAIK I need to:   Shrink the 160GB Ubuntu partition to 80GB Copy it over to the SSD Change fstab to mount the SSD as /   How do I do the second? And what do I need to do about Grub? R : Format the text file output What is the variance of the weighted mixture of two gaussians? Say I have two normal distributions A and B with means $\mu_A$ and $\mu_B$ and variances $\sigma_A$ and $\sigma_B$. I want to take a weighted mixture of these two distributions using weights $p$ and $q$ where $0\le p \le 1$ and $q = 1-p$. I know that the mean of this mixture would be $\mu_{AB} = (p\times\mu_A) + (q\times\mu_B)$.  What would the variance be?    A concrete example would be if I knew the parameters for the distribution of male and female height. If I had a room of people that was 60% male, I could produce the expected mean height for the whole room, but what about the variance? I'm just trying to review basic terminal commands. Having said that, how do I create a text file using the terminal only?