Some proof involving expectation Let $X$ and $Y$ be two random variables with finite second moments and $E(X \mid Y) = Y$ , $E(Y \mid X)= X$. Show that $P(X = Y)= 1$. I tried using conditional expectation, so $E(E(X \mid Y))=E(Y) \Rightarrow E(X)=E(Y)$, but then I'm totally stuck on how to continue. I don't even know how to relate it to probability or using the fact of finite second moment. Any help is appreciated. Thank you.
If $E[X|Y]=Y$ almost surely and $E[Y|X]=X$ almost surely then $X=Y$ almost surely Assume that $X$ and $Y$ are two random variables such that $Y=E[X|Y]$ almost surely and $X= E[Y|X]$ almost surely. Prove that $X=Y$ almost surely.   The hint I was given is to evaluate: $$E[X-Y;X&gt;a,Y\leq a] + E[X-Y;X\leq a,Y\leq a]$$  which I can write as: $$\int_A(X-Y)dP +\int_B(X-Y)dP$$ where $A=\{X&gt;a, Y\leq a\}$ and $B=\{X\leq a,Y\leq a\}$.  But I need some more hints.
How to not count blank pages in frontmatter of two-sided document There does not exist group $G$ such that Aut$(G)\cong \mathbb{Z}_n$ (for odd $n$) Limit of sequence in which each term is defined by the average of preceding two terms We have a sequence of numbers $x_n$ determined by the equality   $$x_n = \frac{x_{n-1} + x_{n-2}}{2}$$  The first and zeroth term are $x_1$ and $x_0$.The following limit must be expressed in terms of $x_0$ and $x_1$ $$\lim_{n\rightarrow\infty} x_n $$  The options are:   A)$\frac{x_0 + 2x_1}{3}$ B)$\frac{2x_0 + 2x_1}{3}$  C)$\frac{2x_0 + 3x_1}{3}$ D)$\frac{2x_0 - 3x_1}{3}$  Since it was a multiple choice exam I plugged $x_0=1$ and $x_1=1$. Which means that all terms of this sequence is $1$,i.e, $$x_n=1, n\in \mathbb{N} $$  From this I concluded that option A was correct.I could not find any way to solve this one hence I resorted to this trick. What is the actual method to find the sequence's limit? Find generator of multiplicative group of $\mathbb{F}_{27}$ I have a finite field $\mathbb{F}_{27}$. I need to find an element of order 13. I know that multiplicative group of this field is cyclic with order 26. So I want to find a generator $g$ of this cyclic group, then $g^2$ will have order 13. Also I know that all elements of this field can be represented by polynomials from $\mathbb{F}_3[t]$, where t is root of irreducible polynom $x^3-x-1$.  Probably we can find generator of $\mathbb{Z}_{26}$ and than build an isomorphism to $\mathbb{F}_{27}\setminus 0$.  Anyway I don't know were to start.  Thanks! Is it possible to increase Output Field Name length beyond 10 characters? When creating a new Column, the default value for the length of the name is 10. Using the Field Calculator via the Model Builder, I want to include the names of regional areas into the columns which has more than 10 characters.  Can the length of the Output Field Name be increased? Can multiple different HTML elements have the same ID if they're different elements? Making a (simple) directory tree Where is the usage of Ubuntu cloud guest image on OpenStack documented? Surface integral of a scalar over a unit cube. Evaluate the following integral $$\iint_S (x+y+z) \, dS$$  where $S$ is the surface of the cube $[0,1] \times [0,1] \times [0,1]$  Honestly, I don't know what to do. All I know is that you have to evaluate the integral for each of the 6 surfaces of the cube. The normals for them I can do and the equations of the planes, but other than that I couldn't do much.  Can someone break it down for me, please?  PS: No divergence theorem yet for me. "like I" or "like me"? In high school we learned to say "than I" and "as I" because you could potentially add an "am" to the end of the sentence.  Examples:  "She is smarter than I."  (Think: "...than I am.") "He is as tall as I."  (Think "...as I am.")  So analogously, shouldn't it be "like I" as well:  "He is sincere, just like I."  (Think: "... like I am.")  But universally, it seems that we use "like me".  Where does this reasoning break down?  Is there history here? Floquet and Bloch's theorems : connection? It is often stated that  and  are equivalent, even the Bloch's theorem is often referred as Floquet-Bloch theorem.  However, it seems quite confusing to me since the former involves a second order differential equation (Schroedinger equation with a periodic potential) while the latter is defined for a first order one.  Can someone clarify this to me? Also it would be nice if I can get references that connect the two. Aligning Vertices Diagonally I am wanting to find the correlation between two variables: a categorical vegetation variable (with categories 1-4)  and a continuous variable of erosion (with values anywhere between 0.2 and 9). I think I would be required to use a non parametric test to find the correlation, but bit of a newby to stats so could anyone suggest the best test to use? I also wanted to know how sample size can affect these tests, ie with a sample size of 100 or 1000? Referring to unidimensional motion, it is obvious that it doesn't always make sense to write the speed as a function of position. Seems to me that this is a necessary condition to derive formulas like:  $$v^2=v_0 ^2 +2\int_{x_0}^{x}a\cdot dx$$  In fact, in the first step of the demonstration (the one I saw, but I think that this step is crucial) it's required to write $a=dv/dt=(dv/dx)(dx/dt)$, that doesn't make sense if $v$ isn't a function of $x$.   When can one rigorously write $v=v(x)$? Cool Text Highlighting in LaTeX LaTeX has soul. ConTeXt, of course, , where you can put sweet hightligher under your text for free!    In the pictured example, you can see highlights that not only highlight, but that do it with cool edges that look so good, you can smell fumes! The fearless author says he'll show how that was done in another post; this one shows only how to lay down a straight, yellow box. But, it is possible to do word-wrapped highlighting with cool-shapped boxes within ConTeXt, presumably.  The trouble for me is that getting into the ConTeXt world is steep climb right now, given the state of documentation and the flux of changes, so I need to continue with LaTeX. But the real problem is that now I can't live without the Organic-Looking Highlights of Awesomeness in that picture!  As I said, Non-Organic-Looking Highlights of the Ordinary are easy with the soul package:  \hl{Some awesome text that word wraps fine.}   It just taunts me on the page with its straight lines. So, if I sacrifice word wrapping, I can use PGF/TiKz and create the following macro:  \newcommand\hl[1]{%     \tikz[baseline,%       decoration={random steps,amplitude=1pt,segment length=15pt},%       outer sep=-15pt, inner sep = 0pt%     ]%    \node[decorate,rectangle,fill=navcolor,anchor=text]{#1\xspace};% }%   Which creates something like this:    Now, my color is too dark and my randomness needs adjusting, but you see that a similar effect could be achieved.  My trouble is that I don't understand TeX enough to dig into the Soul package and add the TiKz/PGF goodness that it so sorely needs. So, I leave it to you guys. Anyone want to tackle this?   Or is it yet another piece of fruit, just a touch out of reach, that shall have to wait until ConTeXt settles down? Should I use static_cast or reinterpret_cast when casting a void* to whatever Both static_cast and reinterpret_cast seem to work fine for casting void* to another pointer type. Is there a good reason to favor one over the other? Why did the Jedi want to bring balance to the Force? During the Prequel trilogy, the Jedi wanted to bring balance to the Force, in accordance with a prophecy.  There seemed to be a general agreement that this was a 'good thing', but a disagreement with the assertion that Anakin was the one to do it.  Why did the Jedi Council believe it was good to bring balance to the Force?  The Republic was mighty (in theory), the Council was established as THE source of Force users, the Sith were thought extinct, and the number of Dark Jedi was apparently very, very low.  It seemed, for all intents and purposes, as if the Force was strongly imbalanced in favor of the Light side...so why did the Council want to 'balance' it? How can I run Windows applications on a MacBook Pro? I have a friend that wants to buy a Mac Book Pro and he has some windows applications that he needs to run. He doesn't want to buy the applications again.  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