Suppose $X$, $Y$ are random variables and $E(X|Y)=Y, E(Y|X)=X$. Is this sufficient to show that $X=Y$? Or at least $X = Y$ 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.
Calculate distance of galaxy 13 billion years ago How can I calculate the distance of a point from Earth now vs 13 billion years ago? The excerpt below claims it is 8 times further now. Can someone provide relevant equations ? How do I copyright my website? How to reduce AWS EBS root volume size? How can I tell if a laptop's video card can be upgraded? First of all, I know that light does indeed travel slower in a medium like air or water, but that's because the photons are bouncing off of the medium's particles and in different directions so the light beam as a whole is slowed down (or at least that's the classical explanation I've read somewhere).  But my question is about light traveling freely in the vacuum. What if an astronaut during a spacewalk switches on a flashlight towards the opposite direction he is moving. Will that light travel slower than normal?  If not, is there another way to make light move slower? For example, can you make a beam of light move so slow that you can visually perceive it being affected by gravity? I am trying to auto-mount my NFS drives, but it doesn't work too well sadly.  What does work is mounting from CLI:  me@client:/$ sudo mount 192.168.178.4:/mountdir /mnt/mountdir me@client:/$ mount -t nfs 192.168.178.4:/mountdir on /mnt/mountdir type nfs (rw,addr=192.168.178.4)   when I try to add mountdir in the fstab, I cannot mount it due to "protocol not supported" (which seems strange as I can mount it)  $ sudo mount films_d3 mount.nfs: Protocol not supported   So there might be something wrong with my fstab entry. This is my current entry:  192.168.178.4:/mountdir /mnt/mountdir  nfs auto,_netdev,noatime,nolock,bg,nfsvers=4,intr,tcp,actimeo=1800 0 0   During startup I also get this error twice: I was trying to fix the mount-from-fstab first, then worry about that, but nevertheless this might be a different problem:  init: idmapd-mounting (/mnt/kazemat/films_d3) main process (445) killed by TERM signal   This seems to be before the eth0 is initialised, while I thought the "_netdev" option would make the entry wait for the network to be up?  can anywone see the mistake in my fstab entry? I didn't like the skill I chosen for the first follower I found. Is it possible to respec his skill later on? What is the difference between Parse() and TryParse()?  int number = int.Parse(textBoxNumber.Text);  // The Try-Parse Method int.TryParse(textBoxNumber.Text, out number);   Is there some form of error-checking like a Try-Catch Block? QR Codes generation from MySQL rows Would the One Ring even work for anyone but Sauron? The temptation of the One Ring, we quickly learn, is always tied up with power: each character's fantasies of possessing it seem to be about what they would do with great power. Even Sam, as I recall, imagines the power to create great huge sweeping gardens. Boromir wants the power to save Gondor, etc.  But does the One Ring actually convey power to anyone but Sauron? It actually seems to diminish its bearers: Bilbo feels "thin" and "stretched", Sméagol becomes the wretched Gollum, Frodo is never quite the same even after it is destroyed. None of them seem more "powerful," even in the abstract way that magic-users in Tolkien operate. No mention is made, that I can recall, of a Ringbearer having greater stature or authority, or of people naturally following them or obeying their commands, while they possess the Ring.  The accepted answer to  implies that a being as powerful as Gandalf could dominate the Ring and break Sauron's hold over it, but even then Tolkien says that "It would have been the master in the end", and his descriptions of a Ringbearing Gandalf sound pretty much like Gandalf is using his own powers, not terribly augmented by the Ring:     [Gandalf] would have remained 'righteous', but self-righteous. He would have continued to rule and order things for 'good', and the benefit of his subjects according to his wisdom (which was and would have remained great).   Now that sounds out of character for a being as humble as Gandalf, but it doesn't actually sound like something he would have been unable to do before. It's a twisted motive, but not really an enhanced power.  Pretty much everyone in Middle-earth agrees that the One Ring will corrupt you, but they all seem to agree that it DOES convey great power even so. Why? Do we ever see any actual, "positive" effects from possessing the One Ring? Arduino Boost Converter. Connecting load makes converter non functional Altium - create a mirror image of a board I want to create an identical motor drive board in the right and left sides.  I just finished layout for the right side and I am wondering what is the easiest way to create the mirror image of this board for the left side?  For example: the connectors on the "top" side would now be on the bottom of the other board.  All components and tracks will also switch sides, layer stack will be inverted, etc.  Is there any command sequence in Altium that will do this without me having to manually re-assign everything to the other layer? Trouble plotting arctan with pgfplots I'm trying to plot arctan(x) with pgfplots but I don't get a good output. Here is my code:   \begin{cfigur}\begin{axis}[      width=160pt,compat=1.5.1,grid style={ultra thin},every axis plot post/.append style={thick},      x tick label style={font=\tiny},y tick label style={font=\tiny},      scale only axis,grid=major,axis lines=middle,      xlabel={$x$},      ylabel={$y$},      xmin=-200,      xmax=200,      domain=-200:210,      ymin=-5.5,      ymax=5.5,      xtick={-150,-100,...,150},      ytick={-5, -4,...,5},      restrict y to domain=-20:20,      legend style={at={(0.5,-0.05)},anchor=north,nodes={right}},  ]  \addplot[mark=none,color=blue, samples=500]{atan(x)};  \addlegendentry{$y = \tan^{-1}x $};  \end{axis}\end{cfigur} Sorting an Array of Objects in PHP In a Specific Order I have two arrays in PHP. The first array ($author_array) is comprised of user_ids in a particular order, like so: (8, 1, 6)  The second array ($user_results) is comprised of an array of objects like so:  Array (     [0] =&gt; stdClass Object         (             [ID] =&gt; 1             [user_login] =&gt; user1         )     [1] =&gt; stdClass Object         (             [ID] =&gt; 6             [user_login] =&gt; user6         )     [2] =&gt; stdClass Object         (             [ID] =&gt; 8             [user_login] =&gt; user8         ) )   I'd like to "sort" the second array so it's in this order, which matches the order of the values in the first array of (8, 1, 6). So it'd look like this:  Array (     [0] =&gt; stdClass Object         (             [ID] =&gt; 8             [user_login] =&gt; user8         )     [1] =&gt; stdClass Object         (             [ID] =&gt; 1             [user_login] =&gt; user1         )     [2] =&gt; stdClass Object         (             [ID] =&gt; 6             [user_login] =&gt; user6         ) )   I'm weak on data structures. How could I do this? :-)  Thanks in advance for your help!  -Bob Ubuntu 17.10 and later: can't change the input switching shortcut to alt+shift When I try to set the default combo for switching to different language layouts in my keyboard from Shift+Super+Space to Alt+Shift I can't.  I used to have this combo since Ubuntu 14 and even before that when I used to have Windows. Now for some stupid reason it does not allow (actually nothing happens its not like I get an error or the combo is used elsewhere). How could I bypass this bug? Why does the base of this slinky not fall immediately to gravity? My guess is tension in the springs is a force > mass*gravity but even then it is dumbfounding. Is there a RSS to the 'Stack Exchange Inbox'? How about to Activity? Just wondering if I'm missing something here. Also, I couldn't find any previous question related to this. "Please ask a new question" encourages question repetition for duplicates I just had a case of a user who re-posted his question because his first question was closed as a duplicate. And, logically, I closed his re-post as a duplicate of his original, asking him to edit his original and explain why it's not a duplicate, so we can reopen it.  This process has worked like that for years now, but suddenly I realized that with the new message, question repetition is even encouraged:     This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.   Clearly, a new user who finds their questions closed will now be happy to just post their question again and again, until it's not a duplicate anymore, instead of just fixing their original post.  We need to be more explicit about this, and show this message to the OP of the question:     If those answers do not fully address your question, please edit your question and explain why it is not a duplicate.   The original message should still be visible to other users and visitors, as they can't fix that question, of course. This morning, I had eggs for breakfast, and I was looking at the pieces of broken shells and thought "What is the surface area of this egg?" The problem is that I have no real idea about how to find the surface area.    I have learned formulas for circles, and I know the equation for an ellipse; however, I don't know how to apply that.   The only idea I can think of is to put an egg on a sheet of paper and trace it, and then measure the outline drawn, and then try to find an equation for that ellipse and rotate that about the $x$-axis.  Now, my problem is how I can find the equation of the ellipse from the graph, and will my tracing method really be the edge of the egg?  Also, can I use the ? Will I have to use some techniques to solve the integral that are not covered in the ?  There has to be a better method for finding the surface area. Please, help me understand how to find the surface area of an egg; i.e., how to use my mathematical knowledge for something other than passing exams. I just picked up a copy of Windows 7 Pro OEM for install on my computer.  Guy at the store said the only real difference between OEM and standard retail box was tech support.  However, the outside of the box has this big warning that says "The software must be installed with an OPK"?  What is the OPK?  Is it really necessary?  The MS site about it seemed to say it was optional.  EDIT:  BTW - I'm just an individual tinkerer, and not a System Builder Company Employee.  I should have made that clear.  I've used sysprep in a professional setting in the past, so I'm somewhat familiar with its concepts.