Since this is a uniform distribution, can we just use symmetry for this, and say that our four numbers will be spaced apart, so they will be on the number line at the following locations: $\frac{1}{5}$, $\frac{2}{5}$, $\frac{3}{5}$, $\frac{4}{5}$ Therefore, the smallest number is expected to be $\frac{1}{5}$.
$X_1, X_2, \ldots, X_n$ are $n$ i.i.d. uniform random variables. Let $Y = \min(X_1, X_2,\ldots, X_n)$. Then, what's the expectation of $Y$(i.e., $E(Y)$)?  I have conducted some simulations by Matlab, and the results show that $E(Y)$ may equal to $\frac{1}{n+1}$. Can anyone give a rigorous proof or some hints? Thanks!
How to move Ubuntu to an SSD My current situation is:   One hard disk Dual boot Ubuntu 11.04 and Windows 7. Partitions:  100MB Windows System thingy 144GB Main Windows 160GB Ubuntu 4GB Swap 12GB System Restore stuff    Now I want to install an 80GB SSD and move Ubuntu to it. AFAIK I need to:   Shrink the 160GB Ubuntu partition to 80GB Copy it over to the SSD Change fstab to mount the SSD as /   How do I do the second? And what do I need to do about Grub? I have looked it up and the resources all talk about schengen visa, travelling with eu national family member etc. so I would like to get a more assuring answer about my situation:   I am a non-EU national with an Italian residence permit. My father is an Italian citizen. I will be travelling to Sofia and Bucharest next month with my girlfriend who is a British citizen. Do I need to get a visa to enter these two cities? What will result in better page ranking in search results: Page views or time spent on page? How to extend my root (/) partition? Till today I was using ubuntu 13.04 in my windows7/ubuntu dual boot. Just now I have upgraded to ubuntu 14.04. My software updater showed an updates available message. I proceeded with  Install Now . Soon I got the following window.    Then I tried sudo apt-get clean as recommended. Nothing happened! Not even a window opened. Teach me deleting the temporary packages.   Deleting temporary files may not be highly helpful in case of further updates and installs for I don't have too much space left in my /. I wanna extend my / partition with the help of the unallocated space in my HD. The following is my Gparted window.       I got an answer . But it wasn't useful for me. I had two choices. But the answer wasn't working for both my chioces.  Choice 1: I may use the 4.25 GiB unallocated space to extend my /dev/sda10. To do so, I must need to move my linux-swap partition. But my Resize/Move option is disabled for the linux-swap partition. Choice 1 failed!   Choice 2: I may use the 14.35 GiB unallocated space instead. In this case I must move more than one partition. /dev/sda4  partition is the first on the list. Even that partition has the same problem. Resize/Move option is disabled. Choice 2 failed!  I am out of choices now.  What shall I do? Help me to extend my / with Gparted. Logarithm as limit lists this as a limit representation of the natural log:  $$\ln{z} = \lim_{\omega \to \infty} \omega(z^{1/\omega} - 1)$$  Is there a quick proof of this? Thanks How to add days to current Date using JavaScript. Does JavaScript have a built in function like .Net's AddDay? Can an Indian travel to India with 50 days before passport expires? Graph with $k$ components. Let $G$ be a simple graph on $n$ vertices. If $G$ has $k$ components and every component is complete prove that: $$|E| \le \frac{(n-k)(n-k+1)}{2}$$ I'm asking for advice. Physical meaning of quantum operators Let's say we have a wavefunction $\psi$ and a measurement operator $\hat A$.  I understand how eigenvalues and eigenvectors of $\hat A$ describe the possible outcomes of the measurement.  I also understand that the average measurement can be computed as $\langle \psi|\hat A|\psi\rangle$.  It's still not clear to me what the direct meaning of $\hat A |\psi\rangle$ is. It is a wavefunction; how does its corresponding quantum state relate (in physical terms) to the original state $\psi$ and the measurement $\hat A$? "whatever" as pure determiner? My dictionaries and references define, and I've always thought of, one of the functions of the word whatever as a "relative determiner."  In a sentence like, "I will help you in whatever way is possible,"  inserting the relative pronoun "that" between "way" and  "is", or replacing "whatever" with "the" - a pure, non-relative determiner - (I reckon) renders it ungrammatical:     *I will help you in whatever way that is possible.   *I will help you in the way is possible.   However, I do often encounter utterances like the first one above, such as:     My hope and intent is to contribute in whatever way that I can.   I wish to be of service to the local communities in whatever way that is possible.   where "that" is used, and thus "whatever" can be replaced by "the" without  making the sentences ungrammatical (if they are grammatical to begin with):     My hope and intent is to contribute in the way that I can.   I wish to be of service to the local communities in the way that is possible.   Is this usage of "that" with "whatever" wrong, though colloquial? Here is a question as phrased in Advanced Linear Algebra (Steven Roman):     Show that a linear operator $\tau$ on a finite-dimensional complex inner product space $V$ is normal if and only if whenever $S$ is an invariant subspace under $\tau$, so is $S^\perp$.   I'm only interested in the $\Rightarrow$ direction. Wikipedia (and ProofWiki)  that uses some kind of computation with the trace. Linear Algebra Done Right chooses an orthonormal basis and then uses the fact that $\Vert \tau v \Vert = \Vert \tau^* v \Vert$.  However, the above  proofs seem a bit too computational to me. Can the following argument work?  (Use $\odot$ for orthogonal direct sum.) Suppose that $\tau$ is normal and write $$V=E_{\lambda_1} \odot \cdots \odot E_{\lambda_k}$$ where the $E_{\lambda_i}$ are the eigenspaces of $\tau$. If $S$ is a $\tau$-invariant subspace then $\tau|_S$ is diagonalizable, so $$S=\widetilde{E}_{\lambda_1} \odot\cdots\odot \widetilde{E}_{\lambda_k}$$ where the $\widetilde{E}_{\lambda_i} \le E_{\lambda_i}$ are the eigenspaces of $\tau|_S$ (which could be trivial). Then choose orthogonal complements $D_i$ such that $E_{\lambda_i} = \widetilde{E}_{\lambda_i} \odot D_i$, and then $$S^\perp = D_1 \odot\cdots\odot D_k.$$ Since each $D_i$ is a subspace of an eigenspace, they are all $\tau$-invariant. propositional calculus problem, how to prove this right or wrong? $A$$\rightarrow$$(B$ $\vee$ $C$ ) , $B$ $\rightarrow$ $C$ $\vDash$ $A$ $\rightarrow$ $D$  I think it's wrong but I have no idea how to prove. Why a mono-atomic crystal layer (2D) can't be stable? According to Peierls and Landau, 2D crystals were thermodynamically unstable. They can't exist! Of course, this theory was disapproved in 2004 (example: graphene).  What is the general definition of stability of a general system?  What is the thermodynamics' stability? Intersection of three vector subspaces We say that two vector subspaces $U,W\subset V$ intersect transversally (written $U\pitchfork W$) if $U+W=V$.  Let $V_1,V_2,V_3$ be vector subspaces of $V$ where $\dim V=n$. We say they intersect normally if $V_i\pitchfork(V_k\cap V_l)$  for $i\ne k,l$. Prove that $V_1,V_2,V_3$ intersect normally if  the following condition holds:  $\operatorname{codim}(V_1\cap V_2\cap V_3)=\operatorname{codim} V_1+\operatorname{codim} V_2+\operatorname{codim} V_3,$   where $\operatorname{codim} V_i=n-\dim V_i$.  How to approach this problem? I tried to order dimensions of $V_i$ but I got that the LHS is $\ge n-\min (\dim V_i,i=1,2,3)$ and the RHS is $\le 3( n-\min (\dim V_i,i=1,2,3))$ which does not lead anywhere. What is the right way to do this? How to force the browser to reload cached CSS and JavaScript files I have noticed that some browsers (in particular, Firefox and ) are very zealous in using cached copies of .css and .js files, even between browser sessions. This leads to a problem when you update one of these files, but the user's browser keeps on using the cached copy. What is the most elegant way of forcing the user's browser to reload the file when it has changed? Ideally, the solution would not force the browser to reload the file on every visit to the page.  I have found  and  suggestion to be useful. It turns out there is a term for this: auto-versioning. I have posted a new answer below which is a combination of my original solution and John's suggestion. Another idea that was suggested by SCdF would be to append a bogus query string to the file. (Some Python code, to automatically use the timestamp as a bogus query string, was .) However, there is some discussion as to whether or not the browser would cache a file with a query string. (Remember, we want the browser to cache the file and use it on future visits. We only want it to fetch the file again when it has changed.) Proving if $d_0$ is the smallest positive integer in $S$ then $d_0 = \gcd(a,b)$ How can we rewrite this optimization problem in standard form? Consider the following problem $$ \begin{array}{crl} &amp;\underset{y}{\max} &amp; f(y)\\ s.t. &amp; y_{1}A_{1}+y_{2}A_{2}+S_{1}&amp;=C_{1},\\  &amp; y_{3}A_{3}+y_{2}A_{4}+y_{5}A_{5}+S_{2}&amp;=C_{2},\\  &amp; y_{3}A_{6}+y_{4}A_{7}+S_{3}&amp;=C_{3},\\  &amp; S_{1},S_{2},S_{3}\succeq0, \end{array} $$ where $y_{i}$ are scalar variables and the matrices $A_{i}$ are symmetric real. How can we rewrite this optimzation problem in the following form  $$ \begin{array}{crl} &amp;\underset{y}{\max} &amp; f(y)\\ s.t. &amp; \sum_{i=1}^{m}y_{i}A_{i}+S&amp;=C,\\   &amp;S&amp;\succeq0, \end{array} $$ where $x_{i}$ are scalar variables and the matrices $B_{i}$ are symmetric real. please? Thanks. What's the meaning of a set to the power of another set? "along the river" vs "alongside the river"? In a question on "along" vs "alongside", I have seen this question and the correct answer on the web site makes me confused.      The question is: I enjoyed my walk _________________ the river.  a) along b) alongside  I selected "...along the river" as the correct answer, but on the web site it is shown as wrong and the correct answer is "alongside".  How can that be? I still do not agree, because I have studied the difference between the two. "along" is just the right preposition for thin and long places like road, path, street, beach etc".   So, why would the correct answer be "alongside" which simply means "next to"? Do you not think "along" seems to be better fitting?  Regards,  Note: I would be more than gratefull, if somebody show it to me in a video, like "a walk along the river" and "a walk alongside the river". Maybe I can understand the how "a walk along the river" and "a walk alongside the river" would take place. I'm getting "jumpy" loadings in rollapply PCA in R. Can I fix it?