conditional probability how to deal with another similar
Conditional probability and venn diagram
How to prove that $E[Y]=E[E[E[Y|X_1, X_2]]]$ ?  PS. I don't see how $E[E(Y|X_{1},X_{2})|X_{1}]=Y[Y|X_{1}]$ and $E[Y]=E[E(Y|X_{1})]$ can be used here. But it feels close. Please help, I'm stuck PPS. Thanks @Deep North. Let's consider the case when X1 is independent of X2 Here is my proof, I would appreciate it if someone could critique it for me:  To prove this statement true, we must proof that the two conditional statements ("If $\mathcal{P}(A)⊆ \mathcal{P}(B)$, then $A⊆B$," and, If $A⊆B$, then $\mathcal{P}(A)⊆ \mathcal{P}(B)$) are true.    Contrapositive of the first statement: If $A \nsubseteq B$, then $\mathcal{P}(A) \nsubseteq \mathcal{P}(B)$  If $A \nsubseteq B$, then there must be some element in $A$, call it $x$, that is not in $B$: $x \in A$, and $x \notin B$. Since $x \in A$, then $\{x\} \in \mathcal{P}(A)$; moreover, since $x \notin B$, then $\{x\} \notin \mathcal{P}(B)$, which proves that, if $A \nsubseteq B$, then $\mathcal{P}(A) \nsubseteq \mathcal{P}(B)$. By proving the contrapositive true, the original proposition must be true.    To prove the second statement true, I would implement nearly the same argument, so that isn't necessary to write. So, does this proof seem correct? Also, was the contrapositive necessary? Or is there another way to prove the initial statement? Short story from many years ago: the ancient, brilliant, cruel race was Earthlings How do I load the contents of a text file into a javascript variable? I have a text file in the root of my web app  and I'd like to load it into a variable in javascript.. in groovy I would do this:  def fileContents = 'http://localhost/foo.txt'.toURL().text; println fileContents;   How can I get a similar result in javascript? We have given isosceles triangle $ABC$ with baseline $AB$. Point $M$ is midpoint of AB. Draw perpendicular line to side $AC$ through $M$ which intersects $AC$ in point $H$. Let $P$ be midpoint of $MH$. Show that lines $BH$ and $CP$ are perpendicular to each other.  I also have "hint": Construct height $BD$ to side $AC$ and look at the triangles $ABD$ and $MCH$.  Analytically there is no problem but I get stuck in geometrical approach to solution. Why does turning Caps Lock off take such a long time to work? This is my biggest issue with Linux and the main reason I don't use it as my main working OS.  Whenever I use caps lock to write something in upper case and disable it in order to keep writing the next letter AFTER I've disabled it is written in upper case.  So, my text comes out weird, as SOMETHING *L*ike this  I've tried Linux Mint, Fedora ... and the issue keeps happening.  Searching Google I've found some very rare persons have the same issue, but I was unable to find any solution.  Is there any way to solve this or am I DOOMED *F*or life?  BTW, whenever I ask this question someone suggest I should learn to use shift, instead of caps lock. I know, that's the right way, but I'm used to use caps lock and, well, it should work. How do I trim leading/trailing whitespace in a standard way? Is there a clean, preferably standard method of trimming leading and trailing whitespace from a string in C? I'd roll my own, but I would think this is a common problem with an equally common solution. My storage was not updated how to adapt the sorting of a custom type `@standard` in using biblatex/biber Spartan 6 DCM unstable clock output The link here refers to the clocking resources of spartan-6 FPGA. I am using the DCM-CLKGEN primitive described in the link, to generate a 8x clock based on an input clock. It works fine as long as the input clock is stable.  But now i sweep the input clock at every 90 us by ±5%. I see that although the DCM does not lose lock, it takes very long (15 to 20 us) to change the output freq after changing the input frequency. This is for me undesirable. Any suggestions?  This question was posted on the xilinx forum and the suggestion was to reset the DCM everytime it loses lock. But the problem is that the DCM is not losing lock  Also, this was earlier posted on stackoverflow and now has been reposted here. if $x+2y+z=4$ and $x,y,z$ are real number. then find maximum value of $xy+yz+zx$  putting $x+z=4-2z$ in $y(x+z)+zx = y(4-2z)+zx = 4y-2yz+zx$  i wan,t be able to  go further,could some help me with this Calculate the area of an irregular cyclic convex polygon Why is energy in a wave proportional to amplitude squared? Is there any shortcut/easy way to switch this system proxy settings? Meaning of the chemical potential for a boson gas My lecturer told me that $\mu$, the , is zero or negative, and in the following example, mathematically it acts as a normalization constant. But is there any physical insight about why boson gas can be zero or negative? I think it is due to the fact the photon gas can pop up from nowhere (i.e. vacuum fluctuation). $$ f_{BE}(\varepsilon)=\dfrac{1}{e^{(\varepsilon-\mu)/(k_B T)}-1}$$ Is it possible to get Inkscape's bucket fill to entirely fill regions marked by lines? I've noticed there's a slight gap between shapes and lines in which bucket-fill is applied (to the enclosed areas found therein) and the colour fill in Inkscape, is there any way of removing this gap (besides the older-fashion manual way by manipulating the borders of the bucket fill, which is very tedious)? how disable resume hard disk from Sleep on Shut Down I'm using Windows 7 , I set my drive to sleep after 10 minutes and it works but, when I shut down windows my hard drive resume from sleep. Why? It shouldn't keep hard drive sleep and shut down the PC ? Comment character counter does not trim, while the server does The observable universe is approximately 13.7 billion years old. But yet it is 80 billion light years across. Isn't this a contradiction? Why is it possible to answer a question twice?