Suppose that $(\Omega, \mathcal E, P)$ is a probability space and $X\colon \Omega \to \mathbb R$ is a RV defined on $\Omega$. Denote as $\mu\colon \mathcal B \to [0,1]$ the probability measure on $\mathbb R$ defined as $$\mu(I)=P(X^{-1}(I)).$$  Suppose that $\mu &lt;&lt; dx$, by Radon-Nikodym theorem there exists $f \colon \mathbb R\to [0,+\infty]$ ($f$ is exactly the PDF of $X$) such that $$\mu(I)=\int_I f(x)dx.$$  Is it correct that, given $g\colon \mathbb R \to \mathbb R$, $$\int_\mathbb R g d\mu=\int _\mathbb R g(x)f(x)dx?$$ And if it is, why?
For random variable $X$ that follows some distribution, $f(x)$ is the probability density function of that distribution if and only if $$\mathbb{E}[\phi(X)] = \int_{-\infty}^\infty \phi(x) f(x)dx$$ for all functions $\phi$.  Context: My professor used this in lecture to demonstrate a way to find the distribution of $cX$ given random variable $X$ that follows a specific distribution.   What is the name of this theorem? Have I missed any qualifications/caveats? I am in particular curious about whether "for all functions" is correct. My professor mentioned "positive, bounded, deterministic functions," but I am not sure what he meant by that. Where can I find a proof of this theorem?
I'm studying for a midterm and need some help with proving summation     $$\sum\limits_{k=0}^n\binom{n}{k}2^k = 3^n$$   using the binomial theorem.  This is what I've been thinking so far:   In the binomial theorem, we set $x = 0$ and $y = 2$, so:  $3^n = (x+y)^n = \sum\limits_{k=0}^n\binom{n}{k}y^k$  $ = \sum\limits_{k=0}^n\binom{n}{k}2^k$  Am I getting this correct so far or completely wrong? Any help would be appreciated. Thanks. Lie algebra and tangent space at the identity are isomorphic Define the Lie algebra $\mathfrak{g}$ of a Lie group G to be the set of all left-invariant vector fields of $G$. I want to prove that   $$          f: \mathfrak{g} \rightarrow T_eG \\              X \mapsto X(e) $$ is a linear isomorphism.  The only thing that I have left to prove is surjectivity.Letting $v\in T_eG$, we set $X\in \mathcal{X}(G)$ such that $\forall x\in G,\,X(x)=L_{x_{*,e}}(v)$ (I still have to show that such an $X$ is indeed a vector field on $G$). We get that  $$ f(X)=X(e)=L_{e_{*,e}}(v)=v $$  where the last equality holds because  $$       L_{e_{*,e}}(v)f = v(f \circ L_e) = v(f \circ id_G) = v(f) $$ for any $f\in C^\infty(G)$. (I am thinking of the tangent vectors as derivations).  Now all we have left to do is prove that such an $X$ is left-invariant, and it is therefore an element of $\mathfrak{g}$, but I do not know how. I tried to prove that  $$            \forall f\in C^\infty(G),\, L_{{x}_*,y}(X(y))f=X(L_x(y))f $$ but with no success. Cooperative game vs Multiplayer game Can any one tell what is difference between cooperative game and multiplayer game? I have a cryptography workshop question I'm having trouble with as follows;  Person A creates a cipher $E_k(m)$ which produces a ciphertext from message $m$ using key $k$. The function inside $E$ is kept secret but the length of $E_k(m)$ is known.  Person B recommends "increasing" security of the cipher by instead doing :   $(E_k(m) \oplus m)\ ||\ (E_k(m) \oplus 1111...11)$      Does this in fact increase security of the cipher or increase new problems.  My thinking is, depending on the function within $E$, xoring the output of the cipher with the plaintext message $m$ could expose the key $k$, meaning the extra complexity is for nothing. Am I on the right track, or am I missing something?  I have tried searching for examples of similar schemes and found nothing (which probably means it's not a good scheme) but I need to justify my answer.   Any steers in the right direction would be greatly appreciated, I'm more than happy to do the research myself just unsure what specifically to look for.   Unfortunately the above context is all I have been provided for this question. How do I force TeX to completely use the 1st column? How do I force TeX to completely use the 1st column before writing to the second column?  Right now I dont have enough content to fill one whole page. So the left and right columns are equally filled, leaving 25% symmetrical empty page at bottom. How do I fill the left column first, and then the right?  \documentclass{article}  \usepackage{multicol} \usepackage[scale=0.9]{geometry} \usepackage{amssymb} \renewcommand{\labelitemi}{\tiny$\blacksquare$} \begin{document} \begin{multicols}{2}      %lots of text goes here  \end{multicols} \end{document} Saving Mathematica kernel state? Which browsers are officially supported, and what else do I need? Which browsers are officially supported by Stack Exchange?  Also, what else is needed in order to use every feature of the site? If $f$ is a concave function and $f(0) \geq 0$ then $f(a+b) \leq f(a) + f(b)$ for all $a,b >0$ Construct a command by putting a string into a tty I managed to do this  echo -n "command" &gt; /dev/tty1  The letters appear, and the cursor moves, but they are "ghosts" – if you hit Enter, nothing happens (they are not in stdin).  Edit:  In the middle of the below screenshot, you see why I see the use of this. (The line with a red caption, right below the line with a yellow caption.) As it is now, you are not really "editing" the note text; you are just asked to write a new text, which will replace the text of the note you are (not really) editing. Thus, I thought it could be remedied by simply pasting the old text into the tty: if the user hits enter, no modification is made. (This program is in Perl/MySQL, but I thought it would be more interesting to ask for a general solution than "how do I do this in Perl".)    Edit 2:  Here is the Perl code, that uses the C code below (works exactly as intended), as well as a new screenshot – hopefully this will clarify things beyond doubt :) Again, look at the middle of the screenshot, where the edit is made to the note text - this time around, the old text is there, for example if you just wanted to fix a typo, you won't have to retype the entire note text.  my $edit_note_text = $edit_note_data[2]; print BOLD, RED, " new text: ", RESET; system("writevt /dev/tty \"$edit_note_text\""); my $new_text = &lt;$in&gt;; $new_text = fix_input($new_text); my $set_text = "UPDATE notes SET note = \"$new_text\" WHERE id = $edit_note_id"; $db-&gt;do($set_text); Shape of the universe? I want to install lubuntu on my current (pre-existing) Ubuntu Install because of it's light-weight footprint.   Can I preserve currently installed apps after installation? How is prototyping done in a professional EE environment? Do you do your prototypes on breadboards, on copper clad boards, Manhattan style, or do you just draw schematics, order PCBs made and assembled and work with them, hacking here and there if necessary? Are amounts of money singular or plural? Are the two sentences correct or is one of them incorrect?     1) Five billion dollars is earmarked for the project.   2) Five billion dollars has been earmarked for the project.   I know dollars are countable. So I think both 'is' and 'has been' are correct here.   Also, do they differ in meaning? Does a US citizen traveling by plane in Mexico (international flight), for example from Mexicali to Mexico City need to have a passport? AMT stock option scenario How do I efficiently iterate over each entry in a Java Map? If I have an object implementing the Map interface in Java and I wish to iterate over every pair contained within it,  what is the most efficient way of going through the map?  Will the ordering of elements depend on the specific map implementation that I have for the interface? How did planets have an elliptical orbit in the first place? I know that planets speed up when getting closer to the sun, but they speed up because they have an elliptical orbit, and they have an elliptical orbit because they speed up. Why did they have an elliptical orbit in the first place? Quantum entanglement faster than speed of light? Bit of an issue here, trying to render this glass scene    Looks fine in the viewport, but when I hit render    Does anyone know the cause of different view port and final render, and how to fix it?  Here's my Pointing the camera in a particular direction programmatically I'm trying to use Python to place a camera. I know the camera's location, and the forward and up vectors for the camera (i.e. the direction it's pointing and its orientation). How do I do this?  I know that it's possible to calculate a raw world matrix from the above information, which I can then apply to the camera, but I was rather hoping not to have to. Is there an easier way?