Let's fix the generic probability space $(\Omega,\Sigma,P)$, and consider the generic random variable $X\colon \Omega \to \mathbb{R}$. The expected value of $X$ (supposed to exist) is then defined as $E(X):=\int_{\Omega} X\,\text{d}P \in [-\infty,+\infty]$, namely using the Lebesgue integral of $X$ with respect to the probability measure $P\colon \Sigma \to [0,1]$. When $X$ is a continuous variable, namely when there exists $f_X\colon \mathbb{R} \to [0,+\infty)$ such that $\forall \,x \in \mathbb{R} \quad F_X(x):=P(X \le x)=\int_{-\infty}^x f_X(t)\,\text{d}t$, we have also that $E(X)=\int_{-\infty}^{+\infty} xf_X(x)\,\text{d}x$. My question is: how can I show that the general definition of $E(X)$ reduces to the last one for continuous random variables? Can you suggest me a textbook or a source where I can find some help? P.S. I have found here  that $\int_{\Omega} X\,\text{d}P=\int_{\mathbb{R}} \text{d}\mu_X$, where $\mu_X\colon \mathcal{B}(\mathbb{R}) \to [0,1] \mid \mu_X(A)=P(X \in A)$ is the pushforward measure of $X$, with $\mathcal{B}(\mathbb{R})$ the Borel $\sigma$-algebra over $\mathbb{R}$. I know also that $F_X'=f_X$ and that $F_X(x)=\mu_X((-\infty,x])$, so maybe $\text{d}\mu_X=\text{d}F_X=f_X\,\text{d}x$, but I seriously think this is totally wrong. Thank you for your time and help!
I recently found out that the expectation of a random variable $X$ in a probability space $(\Omega, \mathcal F,  \mathbb P)$, $\mathbb E(X)$, is just the term used in probability theory for the measure of the function $X$; i.e.:  $$ \mathbb E(X)=\mathbb P(X) $$  or, for a real-valued random variable $X:\Omega\to\mathbb R$:  $$ \mathbb E(X)=\int_\Omega X(x)\mathbb P(dx) $$  The thing is, I am also familiar with the definition of the expectation of a random variable $X:\Omega\to\mathbb R$ with probability distribution function $f$ to be:  $$ \mathbb E (X)=\int_\mathbb R xf(x)dx $$  Here, the integral is with respect to the Lebesgue measure on $\mathbb R$.    We can define the function $f$ in the following way: we have $F(x)=\int_\mathbb R f(x)dx$, where $F(x)=\mathbb P(X^{-1}((-\infty,x]))$.    I can't see any way to show that these two definitions are equal to one another.  The only transformation between integrals with respect to different measures that I know is that $\mu(f^{-1}(g))=\mu(g\circ f)$, which gives that $\mathbb E(g)=\mu_X(g)$, where $\mu_X$ is the image measure $\mathbb P\circ X^{-1}$, but that doesn't seem to help.  Why can we write $\mathbb E(X)=\int_\mathbb R xf(x)dx$?  Why are these two definitions consistent?
Will I die if I lose enough life to kill me from an effect that also heals me? What happens if I dip below 1 life from an effect that will also heal me after the damage is dealt?  For example:   If I bolt myself while I'm at 3 or less with a  on the board? Will I die while at 3 or less if I -2 , targeting myself? Will I die if a creature with my  on it damages me? Will I die if I target myself with enough damage to 'kill' me with a ?   and so on and so forth. You get the idea. I'm pretty sure there's a relatively simple answer that I just don't know about. FireFox 66 Quantum disabled all my extensions at midnight! This morning I woke up to:    and the FireFox help page doesn't really help...  Help!  What should I do now??? I need NoScript and AdBlock Plus to improve my surfing speed!!! 0:-) I want to run a script to simply change the current working directory:  #!/bin/bash cd web/www/project   But, after I run it, the current pwd remains unchanged! How can I do that? Reducing voltage with resistors Are files created in "/storage/emulated/0" supposed to appear in "/storage/sdcard0"? From what I guess, internal storage is available by every user. If emulated storage is a storage location, made for a specific user, are files created in /storage/emulated/0 supposed to appear in /storage/sdcard0? I assume no, because files that are user created should show up in emulated only, otherwise files created by the user would be available for all other users to see. I think my MacBook Pro battery was replaced by Apple's service team. How do I check that? I handed my MacBook Pro (15" Touch bar) to the service centre in order to have the keyboard replaced. Now, when I check the battery cycle count, it shows "1". Before it was at "82".   Also, I'm getting a battery life of only 4.5 - 5 hours now.   Is there a way to find if my battery was replaced or not? Modify obj after import using python I am trying to modify an object after importing it. This post is related to   import bpy full_path_to_file = "C:\\Users\\xyz\\Documents\\test.obj" bpy.ops.import_scene.obj(filepath=full_path_to_file) # create the material mat = bpy.data.materials.new('MaterialName') mat.diffuse_color = (1.0,0.0,1.0) mat.diffuse_shader = 'LAMBERT' mat.diffuse_intensity = 1.0  # get the object obj = bpy.data.objects['Cube']  # get the material mat = bpy.data.materials['MaterialName']  # assign material to object obj.data.materials.append(mat)   I wanted to add a material to the imported object but that is not happening. So I tried to scale it even that is not happening.  I realized that no operation is being performed on the imported objects. But the same operations are executing on the objects that are created (i.e. not imported).  Any help would be appreciated. Which annotation should I use: @IdClass or @EmbeddedId The JPA (Java Persistence API) specification has 2 different ways to specify entity composite keys: @IdClass and @EmbeddedId.   I'm using both annotations on my mapped entities, but it turns out to be a big mess to people who aren't very familiar with JPA.   I want to adopt only one way to specify composite keys. Which one is really the best? Why? Interactive / Real time code compilation for stackoverflow.com. That will improve the answer correctness of the stackoverflow.com.  When adding a new code block, let the user try to compile the code and see the result. It will be useful to demonstrate class designs, algorithms, and logics. How do you get \mathbb{1} to work (characteristic function of a set)? How to handle WndProc messages in WPF? In Windows Forms, I'd just override WndProc, and start handling messages as they came in.  Can someone show me an example of how to achieve the same thing in WPF? Install suspension forks on non-suspension bike How do I set the desktop background on Windows from a script? On X Windows I had a cool 'silent-alarm" reminder script that would change my root window (background) color to solid red, just for a few seconds a few moments before changing it back. Is there a way to do this for Windows XP?  I'm thinking some kind of scheduled task that uses cscript to set registry keys (HKEY_CURRENT_USER\Control Panel\Desktop) . However my attempts don't seem to have any effect. What do I have to signal to read those registry entries and re-draw the desktop? Beamer, Babel and Tikz conflict I am preparing lecture notes in Turkish, and I have a digram for mathematical modelling procedure. But I cannot produce a good output for the following code. When I remove the first three packages, I get my desired output. However, I need to have keep them in my document. Would you help me in this direction?  \documentclass[10pt,notheorems]{beamer}  \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[turkish]{babel}  \usepackage{tikz} \usetikzlibrary{positioning,shapes,arrows}  \hypersetup{bookmarksdepth=4,bookmarksnumbered=true,bookmarksopen=true}  \usetheme{Warsaw} \usecolortheme{seahorse}  \begin{document}  \begin{frame}[fragile]{}{}  \begin{figure}   \centering   \begin{tikzpicture}     \node (rwp) [draw,fill=blue!20,text centered,minimum height=2em,rounded corners] {Ger\c{c}ek ya\c{s}am problemi};     \node (mm) [right=2cm of rwp,draw,fill=purple!20,text centered,minimum height=2em,rounded corners] {Matematiksel model};     \node (mc) [below=2cm of mm,draw,fill=green!20,text centered,minimum height=2em,rounded corners] {Matematiksel \c{c}{\i}kar{\i}m};     \node (rwq) [below=2cm of rwp,draw,fill=red!20,text centered,minimum height=2em,rounded corners] {Ger\c{c}ek ya\c{s}am tahminleri};     \path [draw,-&gt;] (rwp.east) -- node [above] {Form\"{u}lasyon} (mm.west);     \path [draw,-&gt;] (mm.south) -- node [right] {\c{C}\"{o}z\"{u}m} (mc.north);     \path [draw,-&gt;] (mc.west) -- node [below] {Yorum} (rwq.east);     \path [draw,-&gt;] (rwq.north) -- node [left] {Kontrol} (rwp.south);   \end{tikzpicture}   \caption{Modelleme s\"{u}reci}\label{s2ss2fig1} \end{figure}  \end{frame}  \end{document} I'm arriving from New Delhi, India (Indian passport) with JAL at Narita on 11th July at 07:30 and departing for San Francisco with JAL at 00:05 on 12th July from Haneda. Do I need a transit visa as I'll be switching airports and spending a few hours at JAL City Haneda Airport Hotel?  Previously I had connecting flights from Narita to San Francisco and obtained shore passes to relax at Narita Nikko Hotel. Now I'm travelling with my elderly Mom and do not want complications at Narita immigration or later in the day while going from Narita Airport to Haneda airport. Please advise. Prove that $(ma, mb) = |m|(a, b)\ $ [GCD & LCM Distributive Law] This is from a problem seminar and I need help figuring out the solution.  Four bugs, $A,B,C,D$ are initially placed at the corners of a unit square. From a given initial moment, all four crawl simultaneously at one and the same speed $s&gt;0$, $A$ towards $B$, $B$ towards $C$, $C$ towards $D$, and $D$ towards $A$, with each heading at every instant along the line joining it with its target.  Find a system of differential equations describing the trajectory of $A$.  Do the bugs eventually all meet at the center of the square? If so, how long did it take?  My attempt so far: $\def\grad{\mathbf\nabla}$ $\newcommand{\norm}[1]{\left\Vert #1 \right\Vert}$ Call ${\vec x}_1,{\vec x}_2,{\vec x}_3,{\vec x}_4$  the positions in the plane of $A,B,C,D$ respectively. Then $$\grad {\vec x}_1 = \frac{{\vec x}_2-{\vec x}_1}{\norm{{\vec x}_2-{\vec x}_1}}s$$ $$\grad {\vec x}_2 = \frac{{\vec x}_3-{\vec x}_2}{\norm{{\vec x}_3-{\vec x}_2}}s$$  $$\grad {\vec x}_3 = \frac{{\vec x}_4-{\vec x}_3}{\norm{{\vec x}_4-{\vec x}_3}}s$$  $$\grad {\vec x}_4 = \frac{{\vec x}_1-{\vec x}_4}{\norm{{\vec x}_1-{\vec x}_4}}s$$  with ${\vec x}_1(0) = (0,0),\  {\vec x}_2(0) = (0,1),\ {\vec x}_3(0) = (1,1),\ {\vec x}_4(0) = (1,0)$.  I have noticed that the sum of the gradients is 0, but I don't know if that is helpful or not. Can someone help me figure this problem out? Frea is trying to kill me I've been trying to play the Dragonborn DLC for Skyrim on the PS3. In order to advance the story, I need to talk to Frea and her dad. But every time I go near the village they try and murder me for slaughtering the village back when I was like level 15 (I'm level 43 now).  I've tried yielding but they won't accept it and I don't know how to make them stop, all I want to do is finish the story. Does anybody know how to fix this issue? Intersection of kernels and linear dependence of functionals Integral $\int_0^\infty \frac{\arctan(x) dx}{x(1+x^2)}$? How would one evaluate the integral $$\int_0^\infty \frac{\arctan(x) \,dx}{x(1+x^2)}$$?  I was told it had a nice closed form and could have been solved with differentiation under the integral sign; however, I tried to set $$I(\alpha) = \int_0^\infty \frac{\arctan(\alpha x) \,dx}{x(1+x^2)}$$ and got nowhere (the resulting integral was very messy). Is there a much more clever substitution that can be used to tackle the integral?