Let $X$ and $Y$ be square integrable random variables s.t. $E[X|Y]=Y$ and $E[Y|X]=X$. Prove $P(X=Y)=1$.  Furthermore, when the condition changes to $X$ and $Y$ are integrable, show that the conclusion is still true.  So far, I only know $E[X]=E[Y]$, which means $E[X-Y]=0$. So, that means I need to show $var(X-Y)=0$, right? Any hints would be appreciated.
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.
probability of $k$ boxes contain exactly $1$ ball Can an irrational number raised to an irrational power be rational? Can an irrational number raised to an irrational power be rational?  If it can be rational, how can one prove it? All-in-One Event Calendar: Custom Query - Getting each event Instance Prove $\mathbb{Z}$ is not a vector space over a field This is an exercise from Chapter 3 of Golan's linear algebra book.  Problem: Show $\mathbb{Z}$ is not a vector space over a field.  Solution attempt:  Suppose there is a such a field and proceed by contradiction. I will write multiplication $FV$, where $F$ is in the field and $V$ is an element of $\mathbb{Z}$.   First we rule out the case where the field has characteristic 2. We would have   $$0=(1_F+1_F)1=1_F1+1_F1=2$$  a contradiction.  Now, consider the case where the field does not have characteristic 2. Then there is an element $2^{-1}_F$ in the field, and   $1=2_F(2^{-1}_F1)=2^{-1}_F1+2^{-1}_F1$  Now $2^{-1}_F1\in\mathbb{Z}$ as it is an element of the vector space, but there is no element $a\in\mathbb{Z}$ with $2a=1$, so we have a contradiction.  Is this correct? Foobar2000-like music player? Tagging best practices I am wondering what I should do when tagging a question that fits multiple tags, in the particular case were the question might belong to general as well as more specific tags.  Let me give you an example to illustrate this:  Let's say I have a question about ASP.NET MVC 3.0 and I use C# as my main language.  The following tags could apply:   c# .net .net-4.0 asp.net asp.net-4.0 asp.net-mvc asp.net-mvc-3   Tagging with C# might make .NET redundant (I don't think C# is used anywhere else outside of the .NET scope), and tagging with  certainly makes  and  redundant. However, my question would get a better visibility with tags such as ,  and , as probably no one browses a tag as specific as .  How shall I tag my questions? Using more general or more specific tags, or a mix of the two? I am working on a problem and I am confused if exponents can be split up in the manner below.   $$x^{k+2} = x^{k+1}+x^{k+1} = 2x^{k+1} $$  I apologize if this is a simple question I tried to look up exponent rules but I couldn't find exponents + a number. Requesting review of technically inaccurate answers Some answers are plain wrong, or even harmful. Usually these get spotted pretty quickly and either fixed or deleted by the poster, on the basis of comments/votes.  Sometimes though these slip under the radar - the OP ignores the downvotes/comments, or nobody even picks up on it the first time around.  I've always assumed that VLQ flags in the 10K queue for answers that are technically flawed, but otherwise OK were effectively "request for review by somebody knowledgeable in this area". It seems though that .  How should answers like this be handled instead if VLQ is not appropriate? Is there a mechanism for requesting further review? Raising them in chat is sometimes an option, but not always appropriate.   The useful actions that might happen as a result of increased visibility:   Edits by domain experts Downvotes to hide the poor answer Less flawed answers by experts Delete votes by 20K users   Would a "technical issue flag" that pushes to a tab in  /review help? In which step of compilation are comments removed? Examples of the Pigeonhole Principle As most of you might know, the Pigeonhole Principle basically  that      If $n$ items are put into $m$ containers, with $n&gt;m$, then at least one container must contain more than one item   It always surprises me how this trivial - and at the same time powerful - idea might be the key in order to solve extremely complicated math olympiad-problems...  Quick and beautiful solutions are characteristic of pigeonhole problems, which are often a three-part process   Recognize that the problem requires the Pigeonhole Principle Figure out what the pigeons and what the pigeonholes might be After applying the pigeonhole principle, there is often more work to be done   I'll illustrate this with an example I've always liked...     (Example-)Problem: Given a $n\times n$ square, prove that if $5$ points are placed randomly inside the square, then two of them are at most $\frac{n}{\sqrt2}$ units apart.   Step 1: This problem can be solved with the Pigeonhole Principle  Step 2: We divide the $n\times n$ square into four $\frac{n}{2}\times\frac{n}{2}$ squares (pigeonholes). Consequently, at least two points (pigeons) are inside the same $\frac{n}{2}\times\frac{n}{2}$ square.  Step 3: The maximal distance between two points in an $\frac{n}{2}\times\frac{n}{2}$ square is the diagonal, which has the length $\frac{n}{\sqrt2}\qquad\square$  Another problem which can be solved with the Pigeonprinciple is the following:     IMO $1972/1$           Prove that from a set of ten distinct two-digit numbers (in the decimal system), it is possible to select two disjoint subsets whose members have the same sum.      At this point, you might have noticed how useful the Pigeonhole Principle can be, if you know how to recognize and use it.     Question: I would like to work on this amazing principle with my students for a week and was, therefore, gathering problems related to the Pigeonhole Principle with beautiful solutions.   Could you suggest some more? What are the meaning and possible origin of "word!" and "word up"? Use sub and superscript as [optional] parameters for macro I have a div with two images and an h1. All of them need to be vertically aligned within the div, next to each other. One of the images needs to be absolute positioned within the div. What is the CSS needed for this to work on all common browsers? &lt;div id=&quot;header&quot;&gt;   &lt;img src=&quot;..&quot; &gt;&lt;/img&gt;   &lt;h1&gt;testing...&lt;/h1&gt;   &lt;img src=&quot;...&quot;&gt;&lt;/img&gt; &lt;/div&gt; Combining randomForests in R, why are the err.rate, mse and rsq components NULL From ?combine in :     The confusion, err.rate, mse and rsq components (as well as the   corresponding components in the test compnent, if exist) of the   combined object will be NULL.   Why are the err.rate, mse and rsq components for the combined NULL? Is there an efficient way to re-calculate these metrics?  I ask because I would like to figure out a way to use the oob re-sampling method with the parRF model in . 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