$\DeclareMathOperator\E{\mathbb E} \DeclareMathOperator\Var{\mathrm{Var}} \newcommand\R{\mathbb R} \DeclareMathOperator\N{\mathcal N} \DeclareMathOperator\tr{\mathrm{tr}}$Suppose $X \sim \N(\mu, \Sigma)$, where $\mathrm{supp}(X) = \R^n$. We can assume $\Sigma$ to be nonsingular. What is $\E \lVert X \rVert$?  If $\Sigma = \sigma^2 I$, then $\lVert X \rVert / \sigma$ follows a , which has known mean (in terms of a ). That gives in this case $$ \E \lVert X \rVert = \sigma \sqrt{\frac{\pi}{2}} \; L_{{1/2}}^{{(n/2-1)}}\left({\frac {-\lVert \mu \rVert^{2}}{2 \sigma}}\right)\, ;$$ if we further assume $\mu = 0$ then it becomes a chi distribution and $$ \E \lVert X \rVert = \sigma \sqrt {2} \, \frac{\Gamma((n+1)/2)}{\Gamma(n/2)} .$$  In the general case, we have an easy upper bound via Jensen's inequality. Letting $C^T C = \Sigma$ and $Z \sim \N(0, I)$: \begin{align} \left( \E \lVert X \rVert \right)^2 &amp;&lt; \E \lVert X \rVert^2 \\&amp;= \E \lVert \mu + C Z \rVert^2 \\&amp;= \lVert \mu \rVert^2 + 2 \mu^T C \E Z + \E \tr( Z^T C^T C Z ) \\&amp;= \lVert \mu \rVert^2 + \tr( C^T C \,\E Z Z^T ) \\&amp;= \lVert \mu \rVert^2 + \tr( \Sigma ) ,\end{align} where the inequality is strict since we've assumed $\lVert X \rVert$ is not degenerate.  We also have a lower bound in the same way: since $\lVert \cdot \rVert$ is convex, $\E \lVert X \rVert \ge \lVert \E X \rVert = \lVert \mu \rVert$. (I previously had something more complicated here, based on  of the function $z \mapsto \lVert \mu + C z \rVert$, but it gave a worse bound.)  Can we tighten these bounds or, preferably, find an exact expression for $\E \lVert X \rVert$ in the general case? If not, what about the diagonal but non-spherical case, or $\mu = 0$, or other interesting subcases?    There's been some discussion on this site of the distribution of $Y = \lVert X \rVert^2$, particularly  and . If we have the full distribution of $Y = \lVert X \rVert^2$, we may be able to find $\E \sqrt{Y}$. That distribution is gross, though, and analytical answers seem difficult.
Assume you are given two objects whose exact locations are unknown, but are distributed according to normal distributions with known parameters (e.g. $a \sim N(m, s)$ and $b \sim N(v, t))$. We can assume these are both bivariate normals, such that the positions are described by a distribution over $(x,y)$ coordinates (i.e. $m$ and $v$ are vectors containing the expected $(x,y)$ coordinates for $a$ and $b$ respectively). We will also assume the objects are independent.  Does anyone know if the distribution of the squared Euclidean distance between these two objects is a known parametric distribution? Or how to derive the PDF / CDF for this function analytically?
How do I monitor network usage on a Windows server? Transform the NFSA with transition table Transform the NFSA with transition table  $$\begin{array}{ccc} X&amp;a&amp;b\\ A&amp;\{\}&amp;\{F\}\\ B&amp;\{\}&amp;\{\}\\ F&amp;\{\}&amp;\{A,S\}\\ S&amp;\{F\}&amp;\{A,F\} \end{array}$$  and starting state $S$ inta a DFSA and give the transition table of that DFSA. Place the letter $X$ (uppercase) in the top left corner of the matrix. Your states in the first column and inputs in the first row can be in the order of your choice.  The question then offers a $7\times 3$ matrix to be filled in.  Unsure how to transform the NFSA with transition table.  This is my answer but it is incorrect.  $$\begin{array}{ccc} X&amp;a&amp;b\\ \{S\}&amp;\{F\}&amp;\{A,F\}\\ \{F\}&amp;\{\}&amp;\{A,S\}\\ \{A\}&amp;\{\}&amp;\{F\}\\ \{A,F\}&amp;\{\}&amp;\{A,F,S\}\\ \{A,S\}&amp;\{F\}&amp;\{A,F\}\\ \{A,F,S\}&amp;\{F\}&amp;\{A,F,S\} \end{array}$$ Is there a (basic) way to understand why a limit cannot be brought inside of a sum, in certain situations? I have seen that the method of "dominated convergence" implies that sometimes a limit cannot be brought inside of a sum, but is there a more basic way to understand this, preferably without using integrals?   A conceptual or simple answer would be helpful. Thank you! How do I mount .vhd file? I'm using 13.04 Raring amd64 and I need to mount a .vhd file on my ubuntu. I've googled about it and found that virtualbox-fuse package is the answer.  But the package was deleted from apt repository. Is there any way to install the package w/ 13.04? or is there any alternative to virtualbox-fuse? $F$ is a finite field of order $q$. What is the size of $\operatorname{GL}_n(F)$ ?  I am reading Dummit and Foote "Abstract Algebra".  The following formula is given:  $(q^n - 1)(q^n - q)\cdots(q^n - q^{n-1})$.  The case for $n = 1$ is trivial.  I understand that for $n = 2$  the first row of the matrix can be any ordered pair of field elements except for $0,0$. and the second row can be any ordered pair of field elements that is not a multiple of the first row.  So for $n = 2$  there are $(q^n - 1)(q^n - q)$ invertible matrices.  For $n\geq 3$, I cannot seem to understand why  the formula works.  I have looked at Sloane's OEIS A002884.  I have also constructed and stared at a list of all $168$ $3\times 3$ invertible matrices over $GF(2)$.  I would most appreciate a concrete and detailed explanation of how say $(2^3 - 1)(2^3 - 2)(2^3 - 2^2)$ counts these $168$ matrices. Change keyboard layout (English UK) on command line to English US In ZF, how would the structure of the cardinal numbers change by adopting this definition of cardinality? In ZFC, a good way of ordering sets by cardinality is by leveraging the notion of an injection. We define: $$X \lesssim Y \leftrightarrow \mbox{ there exists an injection } X \rightarrow Y.$$  Alternatively, we can leverage the notion of a partial surjection, by which I mean a partial function that hits every point of its codomain at least once.  $$Y \succsim X \leftrightarrow \mbox{ there exists a partial surjection } Y \rightarrow X.$$  Now in ZFC, these approaches are equivalent, in the sense that $$X \lesssim Y \iff Y \succsim X.$$  But if we just consider ZF (that is, we drop choice) then the equivalence breaks down. However, we've still got the forward implication, because the converse of an injection is always a partial surjection.  A specific question. Define that two sets are weakly equipotent iff $X \succsim Y$ and $Y \succsim X$. Question: does weak equipotence imply genuine equipotence? I'm guessing "no."  A general question: In broad terms, how would the structure of the cardinal numbers in ZF be altered by adopting $\succsim$ rather than $\lesssim$ as the fundamental order relation, and taking equality of cardinal numbers to correspond to weak equipotence of sets? I'm guessing this would hide a lot of the (usually very complicated) structure of the cardinal numbers in ZF.  Remark. The order symbols can be written \lesssim and \succsim. finding rational roots of polynomials Could someone please explain me how to apply the Descartes's Criterion? For example , how do I find the rational roots of $ x^3 -x +1$.  I've been looking at some examples, but I get confused. I've had a promising Skype interview with a great professor from a top-level university in US. Our research interests are closely related and he encouraged me to apply BUT he says that all decisions are finally made by admission committee there.  The question is "Is a professor in such a high-level university able to affect admission committee if he really wants a student?" or "Does admission committee decision-making process become easier if a professor really wants an applicant?"  The main point is the difference between my transcript (which is a real catastrophe!) and my research background which is pretty good so the professor's decision might be different from admission committee. With biblatex, the author can be replaced by a dash (or “idem”) in the bibliography. Is it possible to have this replacement also if there are multiple authors, and the first of these multiple authors is the same as the author of the book listed before? (It is somewhat difficult to describe, see the attached example instead, where “Buchautor, Hans-Wilhelm” should be replaced in book3 as well.)   I assume that it should be possible by customising \DeclareNameFormat, but I don’t know how.  Minimal example:  \documentclass{scrartcl} \usepackage[T1]{fontenc} \usepackage[latin1]{inputenc} \usepackage[english]{babel} \usepackage{csquotes} \usepackage{filecontents} \begin{filecontents}{\jobname.bib} @BOOK{book1,   author = {Buchautor, Hans-Wilhelm},   title = {Irgendein Buch},   location = {Buch am Wald},   date = {2000} } @BOOK{book2,   author = {Buchautor, Hans-Wilhelm},   title = {Ein anderes Buch},   location = {Berlin},   date = {2001} } @BOOK{book3,   author = {Buchautor, Hans-Wilhelm and Zweitautor, Zoran},   title = {Noch ein Buch},   location = {London and New York},   date = {2010} } \end{filecontents}  \usepackage[ style=authortitle, backend=bibtex8 ]{biblatex} \bibliography{\jobname}  \renewcommand*{\bibnamedash}{%   \bibstring[\mkbibbold]{idem\thefield{gender}}}  \begin{document} \nocite{*}  \printbibliography \end{document}   Edit 15/02/2011 17:14   I should have added that there was a feature request for biber which pointed back to biblatex. See .  Edit 21/06/2011 9:26  Is this somehow possible now with biblatex 1.4/1.5? I played a bit with uniquename/uniquelist, but I could not get it to work. How to interpret the output of XGBoost importance? I ran a xgboost model. I don't exactly know how to interpret the output of xgb.importance.  What is the meaning of Gain, Cover, and Frequency and how do we interpret them?  Also, what does Split, RealCover, and RealCover% mean? I have some extra parameters   Are there any other parameters that can tell me more about feature importances?  From the R documentation, I have some understanding that Gain is something similar to Information gain and Frequency is number of times a feature is used across all the trees. I have no idea what Cover is.  I ran the example code given in the link (and also tried doing the same on the problem that I am working on), but the split definition given there did not match with the numbers that I calculated.  importance_matrix   Output:             Feature         Gain        Cover    Frequence   1:            xxx 2.276101e-01 0.0618490331 1.913283e-02   2:           xxxx 2.047495e-01 0.1337406946 1.373710e-01   3:           xxxx 1.239551e-01 0.1032614896 1.319798e-01   4:           xxxx 6.269780e-02 0.0431682707 1.098646e-01   5:          xxxxx 6.004842e-02 0.0305611830 1.709108e-02  214:     xxxxxxxxxx 4.599139e-06 0.0001551098 1.147052e-05 215:     xxxxxxxxxx 4.500927e-06 0.0001665320 1.147052e-05 216:   xxxxxxxxxxxx 3.899363e-06 0.0001536857 1.147052e-05 217: xxxxxxxxxxxxxx 3.619348e-06 0.0001808504 1.147052e-05 218:  xxxxxxxxxxxxx 3.429679e-06 0.0001792233 1.147052e-05 What does the Asterisk * mean in Objective-C? I was reading the following article by Mathew Might.  I saw the following the paragraph  What doesn't matter GPA? I don't care if it's 2.0 or 4.0. I won't even look at it. The school you went to? I'll judge you the same whether you went to Nowhere State U or a top-ten school. Transcripts? Never seen one. GREs? Irrelevant. Where you work/worked? Unless it's a research lab, it's not important. I don't think these items have much predictive capacity as to whether or not someone can complete a Ph.D.  Is Mathew Might telling the truth? For me it's very hard believe that GPA, GRE scores and undergraduate school don't matter. I used to think that they play an extremely important role in grad school admission. Statistical interpretation of Maximum Entropy Distribution I have used the principle of maximum entropy to justify the use of several distributions in various settings; however, I have yet to be able to formulate a statistical, as opposed to information-theoretic, interpretation of maximum entropy. In other words, what does maximizing the entropy imply about the statistical properties of the distribution?   Has anyone run across or perhaps discovered yourself a statistical interpretation of max. entropy distributions that does not appeal to information, but only to probabilistic concepts?  As an example of such an interpretation (not necessarily true): "For an interval of arbitrary length L on the domain of the RV (assuming its 1-d continuous for simplicity), the maximum probability that can be contained in this interval is minimized by the maximum entropy distribution."  So, you see there is no talk about "informativeness" or other more philosophical ideas, just probabilistic implications. mongodb $exists always returning 0 Many apps are missing from Ubuntu Software app I tried to enable additional repositories according to the instructions from , however my Ubuntu Software app is still missing a lot of popular apps. Is there anything I'm missing?  Ubuntu 16.10 Ubuntu Software app 3.20.1 Mark a comment as the answer to a question I have created my StackOverflow Careers panel, and when I click 'add a stack overflow account with over 200 reputation' the site decides that I have no StackOverflow account at all, although my current one has over 300. I log in to both with the same StackExchange Account with the same email etc so I'm curious as to why this is happening. Didn't Harry have any other family he could have searched for? I have seen several answers and proofs here at MSE that a self-adjoint operator $T$ on a Hilbert space $H$ where $T^2=T$, implies $T$ being an orthogonal projection operator.   The thought struck me; what would happen if the condition $T^2=T$ is replaced by $T^3=T^2$ or instead by $T^4=T^3$ or why not more generally by $T^n=T^{n+1}$ for some $n\in\mathbb{N}$? Would this still imply $T$ to be an orthogonal projection operator?  I guess someone must have thought of this before me? In that case perhaps there is a reference in the literature for this or perhaps the proof is way simpler than I might expect?