Markov's inequality states that given any nonnegative random variable and $a&gt;0$ then we have:  $$P(X \geq a) \leq \frac{E(X)}{a}$$ At which $a$ is equality supposed to hold?
Find an example where Markov's inequality is tight in the following sense: For each positive integer $a$, find a non-negative random variable $X$ such that $P(X\ge a)=E(X)/a$.   How to do this problem, I am really confused. Also, what is the definition of Markov's inequality?
Avoiding unused variables warnings when using assert() in a Release build Is there such a thing as a restore point in Ubuntu? Windows has this. Basically what I want is to record the state of the system either manually or periodically and then restore to that point if required. For example, before upgrading the distribution, or before installing a certain package.  What would be the easiest way to do this? In Ubuntu 18.04 I'm not able to assign Ctrl+Alt+← or Ctrl+Alt+→ to anything. To be precise I may bind it to something, but it doesn't work when using the keyboard combination, up/down arrows do however work.   .  The strange thing is that the system detects the keyboard combinations when assigning a key, but not when trying to use them. As a result the keyboard combinations don't work in any other programs either. Do MOSFETs have a diode built into them? I have noticed that a MOSFET symbol has a little diode in it (or at least what looks like a diode). Does this mean I do not have to worry about using a diode in a circuit that runs a motor using one? I would have used a diode in order to prevent reverse voltage from turning the motor by hand. Failure to download extra data files - ttf-mscorefonts-installer This is an old problem that deserves a bugfix, but it's getting beyond a joke now and I can't seem to work around it.  I keep getting the dreaded failed To download extra data files error in relation to ttf-mscorefonts-installer:    Repeatedly asking it to try again doesn't help, additionally, repeatedly trying-  sudo rm -rf /var/lib/update-notifier/package-data-downloads/partial/* sudo apt-get --purge --reinstall install ttf-mscorefonts-installer   Isn't working either.  I just get at least one 404 error from one of the sourceforge links that stops it working-  Reading package lists... Done Building dependency tree        Reading state information... Done 0 to upgrade, 0 to newly install, 1 reinstalled, 0 to remove and 20 not to upgrade. Need to get 0 B/29.5 kB of archives. After this operation, 0 B of additional disk space will be used. Preconfiguring packages ... (Reading database ... 352049 files and directories currently installed.) Preparing to unpack .../ttf-mscorefonts-installer_3.4+nmu1ubuntu2_all.deb ... mscorefonts-eula license has already been accepted Unpacking ttf-mscorefonts-installer (3.4+nmu1ubuntu2) over (3.4+nmu1ubuntu2) ... Processing triggers for fontconfig (2.11.94-0ubuntu1.1) ... Processing triggers for update-notifier-common (3.168.2) ... ttf-mscorefonts-installer: processing... ttf-mscorefonts-installer: downloading http://downloads.sourceforge.net/corefonts/andale32.exe Get:1 http://downloads.sourceforge.net/corefonts/andale32.exe [198 kB] Fetched 198 kB in 0s (339 kB/s)                                                 W: Can't drop privileges for downloading as file '/var/lib/update-notifier/package-data-downloads/partial/andale32.exe' couldn't be accessed by user '_apt'. - pkgAcquire::Run (13: Permission denied) ttf-mscorefonts-installer: downloading http://downloads.sourceforge.net/corefonts/arial32.exe Get:1 http://downloads.sourceforge.net/corefonts/arial32.exe [554 kB] Fetched 554 kB in 0s (575 kB/s)                                                 W: Can't drop privileges for downloading as file '/var/lib/update-notifier/package-data-downloads/partial/arial32.exe' couldn't be accessed by user '_apt'. - pkgAcquire::Run (13: Permission denied) ttf-mscorefonts-installer: downloading http://downloads.sourceforge.net/corefonts/arialb32.exe Get:1 http://downloads.sourceforge.net/corefonts/arialb32.exe [168 kB] Fetched 168 kB in 0s (306 kB/s)                                                 W: Can't drop privileges for downloading as file '/var/lib/update-notifier/package-data-downloads/partial/arialb32.exe' couldn't be accessed by user '_apt'. - pkgAcquire::Run (13: Permission denied) ttf-mscorefonts-installer: downloading http://downloads.sourceforge.net/corefonts/comic32.exe Err:1 http://downloads.sourceforge.net/corefonts/comic32.exe   404  Not Found W: Can't drop privileges for downloading as file '/var/lib/update-notifier/package-data-downloads/partial/comic32.exe' couldn't be accessed by user '_apt'. - pkgAcquire::Run (13: Permission denied) E: Failed to fetch https://netix.dl.sourceforge.net/project/corefonts/the fonts/final/comic32.exe  404  Not Found  E: Download Failed Setting up ttf-mscorefonts-installer (3.4+nmu1ubuntu2) ...   The specific point at which it fails to download changed - it seems to be a sourceforge reliability problem, but this has now been going on weeks and results in annoying popups as above all the time.  How do we get this fixed and/or how can I work around it for now. Is there any simple method to calculate $\sqrt x$ without using logarithm Suppose that we don't know logarithm, then how we would able to calculate $\sqrt x$, where $x$ is a real number? More generally, is there any algorithm to calculate $\sqrt [ n ]{ x } $ without using logarithm? More simple techniques would be nice.  Here is a simple technique used to approximate square roots by :  For example: $\sqrt {78}\approx 8\frac { 14 }{ 17 } $ , where $8$ is the nearest integer root of $78$, $14 = 78 - 8^2$, $17 = 2 \times 8 + 1$.  if $n=2^k$ we can use the method above.   For example, for $k=2$ Lets calculate $\sqrt [ 4 ]{ 136 } $: $$\sqrt [ 4 ]{ 136 } =\sqrt { \sqrt { 136 }  } \approx \sqrt { 11\frac { 136-{ 11 }^{ 2 } }{ 11\times 2+1 }  } =\sqrt { 11\frac { 15 }{ 23 }  } \\ \sqrt { 11\frac { 15 }{ 23 }  } \approx 3\frac { 11\frac { 15 }{ 23 } -{ 3 }^{ 2 } }{ 3\times 2+1 } =\frac { 544 }{ 161 } =3.38\\$$ The exact result is$$ \sqrt [ 4 ]{ 136 } =3.4149\cdots$$ The method approximates well, but it is working for only $n=2^k$ as I know. When at home I use my 13" MacBook Pro hooked up to an 27" external monitor. When working on my 27" screen I prefer to have a visible dock at all times. I have enough screen real estate to afford this luxury.  When I'm on the road working on my 13" screen I prefer to automatically hide the dock as it really gets in the way when screen real estate is limited.  I now manually switch between enabling and disabling hiding (using ⌥⌘D) but I would prefer this to happen automatically. Can I enable hiding the dock on my MacBook monitor and disable hiding the dock on my external monitor? Render just part of a final image from Cycles? I'm using an older Nvidia chip that hates running viewport renders, and I need to check if changes to the materials on a few objects not far from the center of my scene are (I'm rendering a floorplan with an orthographic camera from above, so I'm looking down on objects and want to check if the displacement materials are working correctly).  Is there any way to select a bit of the image and have it render first or without having to render the rest of the image? I'm aware that you can tell the image to render from an edge, but my stuff's about equidistant from either side of the scene. What should we do about Stack Overflow-specific questions on Meta SE? Proof that $n^2 < 2^n$ How do I prove the following statement by induction?  $$n^2 \lt 2^n$$  $P(n)$ is the statement $n^2 \lt 2^n$  Claim: For all $n \gt k$, where $k$ is any integer, $P(n)$  (since $k$ is any integer, I assume I have to prove this for positive and negative integers)  So let base case be $P(1)$, and I have to prove $P(n)$ for $n \ge 1$ and $n \lt 1$  $P(1)$ is $1^2 \lt 2^1$ which is clearly true.  Induction hypothesis: $k^2 \lt 2^k$  Inductive step $(k+1)^2 \lt 2^{(k+1)}$  $(k+1)^2 = k^2 + 2k + 1$  $k^2 + 2k + 1 \lt 2^k + 2k + 1$ by inductive hypothesis  Not sure how to proceed. Is my previous intuition that I have to prove this for $n \ge 1$ and $n \lt 1$ correct? I've been trying to wrap my head around the basic concepts of category theory, and I thought I would attempt to illustrate what I understand with the category of sets, probably the easiest example. Particularly, I've been trying to prove that $id_A$ (the identity morphism on $A$, for all $A \in Obj(\mathbf{Set})$) is $1_A \colon A \rightarrow A, x \mapsto x$.  This is a very intuitive and reasonable statement, and it's trivial to prove that $1_A$ is indeed an identity morphism on $A$, and I suppose uniqueness of $id_A$ can be demonstrated analogously to uniqueness of the identity element in a monoid (considering the subcategory which has $A$ as its only object, and endofunctions on $A$ as its only morphisms).  In this manner, it is not hard to prove that the proposition in the title is true, but this demonstration requires to make an assumption or guess as to what could $id_A$ be. Specifically, the scheme of the proof is: assume $id_A$ = $1_A$, see that it works with the definition of an identity morphism, show that the identity morphism is unique, and, in conclusion, $id_A$ can only be $1_A$. What I'm looking for, nonetheless, is a somehow more direct proof, that doesn't assume $id_A$ = $1_A$ at the start. I want to place myself in a state of little or no knowledge about sets and functions, and under this assumption, why would I assume $id_A$ = $1_A$ at first? Why not try with $id_\mathbb{Z}$ = $f \colon \mathbb{Z} \to \mathbb{Z}, x \mapsto x^2 + 1$, for example? It wouldn't work, but I don't have any reason to think that $1_\mathbb{Z}$ is a better guess for $id_\mathbb{Z}$.  I suppose that the proof for which I'm asking would work for categories of sets with additional structure, and probably for posets as well, although I'm not clear as to what modifications it would require to work.  Thanks. Calling a method in java and returning beer cost How can I automatically add some skeleton code when creating a new file with vim When creating a new file with vim, I would like to automatically add some skeleton code.  For example, when creating a new xml file, I would like to add the first line:    &lt;?xml version="1.0"?&gt;   Or when creating an html file, I would like to add:  &lt;!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"&gt; &lt;html&gt;   &lt;head&gt;     &lt;title&gt;&lt;/title&gt;   &lt;/head&gt;   &lt;body&gt;   &lt;/body&gt; &lt;/html&gt; Change default code page of Windows console to UTF-8 Currently I'm running Windows 7 x64 and usually I want all console tools to work with UTF-8 rather than with default code page 850.  Running chcp 65001 in the command prompt prior to use of any tools helps but is there any way to set is as default code page?  Update:  Changing HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\Nls\CodePage\OEMCP value to 65001 appear to make the system unable to boot in my case.  Proposed change of HKEY_LOCAL_MACHINE\Software\Microsoft\Command Processor\Autorun to @chcp 65001&gt;nul served just well for my purpose. (thanks to ) Evaluate $\lim_{n\rightarrow\infty} \int_0^1 f(x^n) dx$ Let $f$ be a real, continuous function defined for all $0\leq x\leq 1$ such that $f(0)=1$, $f(1/2)=2$, and $f(1)=3$. Show that  $$\lim_{n\rightarrow\infty} \int_0^1 f(x^n) dx$$  exists and compute the limit.  Attempt:   Since $f$ is real-valued continuous on $0\leq x\leq 1$ and the boundaries $f(0)=1$ and $f(1)=3$, $f$ is bounded on the interval  and the integral $\displaystyle\int_0^1 f(x^n)dx$ exists for any positive $n$. Thus, we can interchange the limit and the integral and compose the limit, \begin{align*} \lim_{n\rightarrow\infty} \int_0^1 f(x^n)dx&amp;=\int_0^1\lim_{n\rightarrow\infty}f(x^n)dx\\ &amp;=\int_0^1 f\left(\lim_{n\rightarrow\infty} x^n \right)dx\\ &amp;=\int_0^1 f(0)dx=1. \end{align*}  Questions: So, my first question concerns whether interchanging the limit and the integral is correct. It seems it would be justified by the dominated convergence theorem, where $f(x^n)$ is dominated by a function $g(x)=\displaystyle\sup_{0\leq x\leq 1} f(x^n)$ and $f_n(x^n)$, on $0\leq x\leq 1$, converges pointwise to a function $f$ that takes value 1 for $x\neq 1$ and 3 for $x=1$.  My next question is whether the interchange between the limit and composition of the function is allowed. Since the function is continuous and the limit exists at that point, it seems the interchange would be justified. Proving that the triangle inequality holds for a metric on $\mathbb{C}$ Show that $(X,d)$ is a metric space where $X =\Bbb C $ and the distance function is defined as:   $$d(x,y) = \frac {2|x-y|}{\sqrt {1+|x|^2} +  \sqrt {1 + |y|^2}}, \text{ for } x,y \in \Bbb C.$$   I have done the proof of the first two propositions for being a metric, but I'm having a problem in proving the triangle inequality. I have a finite $p$-group $G$ and a normal subgroup $N$ which is not the trivial subgroup. I am asked to show that $|N \cap Z(G)| &gt; 1$.  There has been a similar question on MSE here:  Unfortunately, I cannot comment on other user's questions yet, so I made my own question.  I know that the trick is to use conjugation but I can't follow the logic. Can someone elaborate? Especially on the last part of the top answer. Optimize table layout MySQL 5.5.16 allows anonymous connections Apparently, I can connect to MySQL (at least from my localhost) without having to provide any username or password. I can even put anything as username, as long as the password is empty. I have access to the information_schema and test databases. I can create tables. If I run SELECT USER() it returns the proper value randomstring@localhost, where the random string is whatever I used for username when I connected.  The mysql.user table contains root@localhost, root@127.0.0.1, @localhost and pma@localhost. I manually set encrypted passwords to all the records (UPDATE user SET password=...) and only root has any privileges, the other two records have "N" for all privileges. That record with the empty username looks suspicious, but I don't think it has anything to do with this and I think it was there from the beginning.  To make matters worse, if I add a user with a password, I can only connect using empty string as password (just like if I put anything as username). If I try to connect using the username I created and password I supplied when creating it (CREATE USER .. IDENTIFIED BY ..) it doesn't work.  Does anyone know what is wrong and how I can fix this?  Note: I am using MySQL 5.5.16 for Windows, provided by the XAMPP package. Trailing characters stripped from titles bypass length minimum