Are there specific conditions under which the following is true? E.g., certain distributions, positive RV?  $$\mathbb{E}\left[\dfrac{1}{\sum X_{i}}\right] = \dfrac{1}{\mathbb{E}\left[\sum X_{i}\right]}$$
How to construct an example of a probability distribution for which $\mathbb{E}\left(\frac{1}{X}\right)=\frac{1}{\mathbb{E}(X)}$ holds, assuming $\mathbb{P}(X\ne0)=1$?    The inequality which follows from Jensen's inequality for a positive-valued RV $X$ is like $\mathbb{E}\left(\frac{1}{X}\right)\ge\frac{1}{\mathbb{E}(X)}$ (the reverse inequality if $X&lt;0$). This is because the mapping $x\mapsto\frac{1}{x}$ is convex for $x&gt;0$ and concave for $x&lt;0$. Following the equality condition in Jensen's inequality, I guess the distribution has to be degenerate for the required equality to hold. A trivial case where the equality holds is of course if $X=1$ a.e. Here is an example I found in a problem book: Consider a discrete random variable $X$ such that $\mathbb{P}(X=-1)=\frac{1}{9}, \mathbb{P}(X=\frac{1}{2})=\mathbb{P}(X=2)=\frac{4}{9}$. It is then easily verified that $\mathbb{E}\left(\frac{1}{X}\right)=\frac{1}{\mathbb{E}(X)}=1$.   This example shows that $X$ need not be positive (or negative) a.e. for the equality in the title to hold. The distribution here is not degenerate either.  How do I construct an example, possibly like the one I found in the book? Is there any motivation?
I need to make a converter to go from RS422 full duplex to my Arduino. I have found lots of different chips which I think can do this, but am slightly confused by the terminology. The RS422 device is a laser distance sensor ().  It has RX+, RX-, TX+, TX-. So I have two lines for input and two lines for output.  I could use two RS485 line drivers to achieve the full duplex comms which I need.  There are a lot of chips to do that.  But then on my Arduino (clone: chipkit max32) would I need to use two UART channels? And then alternate which one I am sending each byte on?  The device uses 4 byte instructions. So if I wanted to send an instruction would I send Serial1(firstByte) then Serial2(secondByte), Serial1(thirdByte), Serial2(fourthByte)?   And then receive data the same way? I feel like I am missing something and can do this on one UART channel but am not sure how. What does "...from nonmagical attacks not made with silvered weapons" mean? SSD drives lifetime Some years ago when I was looking to buy an SSD drive, I remember reading about how SSD drives had low lifetime with frequent read/writes, so I backed off.  Now I see this in Specifications:  MTTF: 1,000,000 Hrs    That's 114 years (ok, it could be a few years less, but still WOW)    What has changed since then? What is your personal experience about this technology? Theorem. Let $(X,\mathcal B,\mu)$ a finite measure space, where $\mu$ is a positive measure. Let $\mathcal A\subset \mathcal B$ an algebra generating $\cal B$. Then for all $B\in\cal B$ and $\varepsilon&gt;0$, we can find $A\in\cal A$ such that $$\mu(A\Delta B)=\mu(A\cup B)-\mu(A\cap B)&lt;\varepsilon.$$  I don't think there is a proof in this site. It's a useful result for several reasons:  We know what the algebra generated by a collection of sets is, but not what the generated $\sigma$-algebra is. The map $\rho\colon \cal B\times\cal B\to \Bbb R_+$, $\rho(A,A')=\mu(A\Delta A')$ gives a pseudo-metric on $\cal B$. This makes a link between generating for an algebra and dense for the pseudo-metric. We say that a $\sigma$-algebra is separable if it's generated by a countable class of sets. In this case, the algebra generated by this class is countable. An with the mentioned result, we can show that $L^p(\mu)$ is separable for $1\leq p&lt;\infty$, which makes a link between the two notions. In ergodic theory, we have to test mixing conditions only an a generating algebra, not on all the $\sigma$-algebra. Teams of users rails app Canonical question: One has a Windows 10 installed with display language A, but uses keyboard layout B. What does one need to do, to stop Windows from automatically adding an unwanted keyboard layout from language A to one's system?  My example: I have a Windows 10 in English and I'm using a Slovenian keyboard layout. And this is working just fine, until I run a certain program that Windows ever so intelligently thinks needs an US keyboard layout. Windows adds this unwanted layout automatically, switches to it and doesn't remove it afterwards. The new layout isn't even listed in installed layouts. The only way to remove it after it appears, is to manually add it as a keyboard layout and remove it.  It seems that's a common problem for many people but after many hours of googling, I just couldn't find a good solution for it... It would be perfectly fine with me if I could just completely remove the US keyboard layout from my system, but if I do that, the keyboard stops working in some programs altogether.  Programs that I or other people have this problem with: ICQ, some versions of Microsoft Office (lol), games based on Source engine, some versions of UltraVNC, Adobe CC Suite, and the list goes on. Equal simple field extensions? I have a question about simple field extensions.  For a field $F$, if $[F(a):F]$ is odd, then why is $F(a)=F(a^2)$? How should a family with young children apply for UK visas? We are sponsoring a family to visit us in UK as friends from Madagascar for two weeks. They have a 6 year old and a 16 month old.   Do the kids need their own visas, and are they the standard visitor visa or a special child visa?   Also, when I try and apply on the website I can only apply for one person so how do I apply for all the individuals in the family?   I have looked on the Immigration website and can't find the answers I need. Is it possible to configure Windows 7 to use YYYY-MM-DD instead of DD/MM/YYYY as the date format? How? Where? Finding the dual basis for the space of polynomials of degree $\leq$ 2 limit: $u_n=\frac{1+2^2+3^3+...+n^n}{n^n}$. Let $u_n=\dfrac{1+2^2+3^3+...+n^n}{n^n}$. Prove that : $$\lim u_n=1$$ I tried to prove that $u_n&lt; something$ have limit  is 1 ; but I can't By default after a new user signs up, a redirect to the home page happens.  But in such case I want to redirect such user to another page like node/35.  In this node/35 I then want to display some content. Define additional math operators to be typeset in roman My table is like this:  Sitecode    Month  Amount --------    -----  ------ XX          Jan     1000 XX          Jan     3000 XX          Apr     3000 XX          Apr     1000   What I want is, to show the result something like this:  Sitecode    MonthJAN   MonthAPR --------    --------   -------- XX          4000       4000 Copy-paste functionality stops working after about an hour after each reboot on a Windows 7 64-bit system. Running Google Chrome (with gmail and few other tabs open like Calendar, Reader), MS Outlook (which I don't think has anything to do with the problem — I saw it when outlook was off as well), iTunes (9.1.1.12 if it matters).  Would appreciate hints where and what for to look in a registry, and ideas for possible fix.  That is not an Internet Explorer problem (I don't even run it) — it happens in all applications.  Neither Ctrl-C/Ctrl-V nor context menu right-click Copy-Paste (actually, nothing happens on Copy, so there is nothing in a clipboard to Paste) are working. Drag-and-Drop (where supported) continues working though. I use Chrome across 4 different workstations and I have my extensions synced across all of them.  The problem I have is that one of the extensions () works great if you've got an actual mouse, but on a trackpad on Ubuntu, it just gets in the way.  If I disable it on the Ubuntu machine, it gets disabled on all Chrome installations due to its .  Q: Can I selectively disable an extension on just a single machine? I need backup in pressing my case that the phrase “choose from one of four options” is grammatically incorrect. Is there some resource that can prove my case, that the incorrect phrase should be replaced with one of the two following ones?        Choose one of four options   Choose from four options How to urlencode data for curl command? What is the simplest SQL Query to find the second largest value? What is the simplest SQL query to find the second largest integer value in a specific column?   There are maybe duplicate values in the column. "Do you want to continue[Y/n]?" Why the upper case? I am new to Ubuntu and some little thing start to bug me. Such as when trying to install a program, Ubuntu asks,     Do you want to continue[Y/n]?   What is bugging me is why the letter "Y" is upper case and "n" is lower case. Do they have any significant reason why one is upper and the latter is lower? Whatever case I enter it doesn't change the outcome: it either continues or not. Are they trying to notify us that we can enter both and it won't change anything?