Suppose that $X$ is a positive random variable, with expectation $\mu$ and variance $\sigma^2$. Prove, for all $\lambda\in(0,1)$, we always have $$P(X&gt;\lambda \mu) \geqslant (1-\lambda)^2 \frac{\mu^2}{\mathbb{E}[X^2]}$$ Can someone sort out the proof using Chebyshev's inequality? Thanks very much.
If Z is a square integrable random variable such that $E[Z]&gt;0$, then prove that  $$P(Z\geq k) \geq \frac{(E(Z)-k)^2}{E(Z^2)} \forall k \in [0,E(Z)]. $$  This is a homework question and I don't even understand why the inequality is reversed comparing to Markov/Chebychev inequality. Should Cauchy-Schwartz inequality be used? If yes, then how to begin? Thanks a ton!
How do you name your ViewModel classes? Multiple regression or partial correlation coefficient? And relations between the two I don't even know if this question makes sense, but what is the difference between multiple regression and partial correlation (apart from the obvious differences between correlation and regression, which is not what I am aiming at)?  I want to figure out the following: I have two independent variables ($x_1$, $x_2$) and one dependent variable ($y$). Now individually the independent variables are not correlated with the dependent variable. But for a given $x_1$ $y$ decreases when $x_2$ decreases. So do I analyze that by means of multiple regression or partial correlation?  edit to hopefully improve my question: I am trying to understand the difference between multiple regression and partial correlation. So, when $y$ decreases for a given $x_1$ when $x_2$ decreases, is that due to the combined effect of $x_1$ and $x_2$ on $y$ (multiple regression) or is it due to removing the effect of $x_1$ (partial correlation)? When dragging an object, there is a greater start-up force than the force it takes to keep it moving. Why is this? Why are there two different values for friction? Is there a clipboard changed or updated event that i can access through C#? What is the best way to execute a script when entering into a directory? When I move into a new directory I would like bash to execute the projectSettings.bash script much like RVM does. Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ have the same number of elements for each order, but $G$ is not isomorphic to $H$? I suspect that there is, but I haven't thought of one. Each spell has it's own casting time, e.g.  takes 1 full round and  takes a swift action.   On the other hand,      Drinking a potion or using an oil is a standard action    as      In many ways, they behave like spells in potion form, and as such their effects can be dispelled by effects like dispel magic using the alchemist's level as the caster level. Unlike potions, though, extracts can have powerful effects and duplicate spells that a potion normally could not.   So how long does consuming an extract take? I'm inclined to believe that it takes just as long as the spell does to cast (e.g. a full round to Enlarge Person, only a swift for Burst of Speed) but I'd love to know if this is ever clarified in the rules or a FAQ that I've missed. $SL_n(\mathbb R)$ is a subgroup of kernel of $f\in$ Hom$(GL_n(\mathbb R), A)$, $A$ is an abelian group. Try to prove the following proposition.     $SL_n(\mathbb R)$ is a subgroup of kernel of $f\in$ Hom($GL_n(\mathbb R), A)$, where $A$ is an abelian group.   My first attempt is to show that $SL_n(\mathbb R)$ is in form of $ABA^{-1}B^{-1}$, then it's clear that $ABA^{-1}B^{-1}$ is in $ker(f)$. However, there exists an elementary row reduction matrix, i.e. $E_{ij}(m)$ adding m times $j$-th row to $i$-th row. Then $E_{ij}(m)\in SL_n(\mathbb R)$. Now, it is possible that $E_{ij}(m)$ is not in $ker(f)$. Then I try to find any relation between $\mathbb R^{\times}$ and $A$, but, except knowing they are both abelian groups, I cannot figure out what else connects those two group. Could you give me some hints? How to model the bottle threads? The picture show the threads I want to create on the highlighted object. Problem is I don't know how and tutorials really haven't helped much as most of them seem to be about creating a bolt. So I was hoping I could get some help on how to do this efficiently.While keeping the top flat like a water bottle. The picture shows the threads I'm after. set only lower bound of a limit for ggplot Array List count to users We're planning to upgrade an existing farm from 2007 to 2013. I learned that the best way and maybe the only way to achieve this is by migrating in two steps. 2007 to 2010 and 2010 to 2013. The actual farm is a sql-server(2008r2 64 bit) and a sperate sharepoint-server(web-frontend, etc....) I want to keep the sql-server as it is, if possible and we want to install a new 64bit host for sharepoint 2013. We will design a new branding for 2013 afterwards, so we need to keep the content, but not the layout. There's so much about migration available on the web, but Can I find something like best practises? How to prove that the center of the dihedral group $D_{2n}$ is $\{1,r^{n}\}$ and the center of $D_{2n-1}$ is $\{1\}$?  I don't know how to prove it in this general case. Where does Drupal store its content? I have been currently learning Drupal while trying to duplicate a Drupal application into CakePHP.  I have been looking at the  database tables to know where data is being stored so I can safely migrate any data or study how it is working.  I have researched and have come across questions like:     So far, I have only seen the title in my Drupal (7) installation.  What I have seen in my node revisions table for a recipe content type, which includes fields like, field_ingredients, field_instructions, field_description, field_preptime, field_cookingtime, just the title of the recipe. This is a sample record on my node_revision table.  nid   |  vid  |  title         |  log   | timestamp  | status | comment | promote  | sticky ------------------------------------------------------------------------------------------- 1     |  1    |  Awesome Sauce |        | 1369998600 | 1      | 2       | 1        | 0   As you can see I can only see the title. Where are the other field contents? Counting the number of propagating degrees of freedom in Lorenz Gauge Electrodynamics How do I definitively show that there are only two propagating degrees of freedom in the Lorenz Gauge $\partial_\mu A^\mu=0$ in classical electrodynamics.  I need an clear argument that   involves the equations of motion for just the potentials $A^0$ and $\mathbf{A}$, and not the electric and magnetic fields.   includes sources $\rho$ and $\mathbf{J}$ in the equations of motion.  This is to justify the assertion that there are degrees of freedom that decouple from the rest of system. does not critically rely on quantum field theoretic arguments (although any supplementary remarks are welcome).     To illustrate the level of clarity I expect, I provide an argument in the Coulomb gauge $\nabla\cdot \mathbf{A} = 0$:  Of the four field degrees of freedom, the gauge condition $\nabla\cdot \mathbf{A} = 0$ removes one degree of freedom (the longitudinally polarized EM waves).  To show that among the remaining three degrees of freedom only two are propagating, consider the field equations of motion in the Coulomb gauge:  \begin{align} \nabla^2 A^0 &amp;= -\rho/\epsilon_0,\\ \big[\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2\big]\mathbf{A}&amp;= \mu_0 \mathbf{J}-\nabla \frac{1}{c^2}\frac{\partial}{\partial t}A^0. \end{align}  The first equation is NOT a wave equation for $A^0$, and thus does not propagate.  The final equation IS a wave equation, and describes the propagation of two degrees of freedom (Gauss' law in first eq. can be solved, and then inserted into the second equation to show that $\mathbf{A}$ only couples to the Solenoidal part of the current). Seeking options for Spatial ETL (Extract, Transform, Load)? I am interested in the pros and cons of various spatial ETL (extract, transform, load) tools.  If you have used the items listed here (or add your own), I seek your opinions and experiences.  In particular I would like to see usability comparisons of:    (proprietary)  (proprietary, subset of FME features)  (open source)  (open source)  (I don't know that it is truly ETL, but more of a file translation toolset) Any additional lot listed   There is no need to give a review of ALL software mentioned.  If you are experienced with even one then that will be very beneficial in making a decision about which direction to go.  Example: I am looking to create a schema conversion function that will allow me to select the input layer, create a translation, and output to a new, pre-defined schema.  Optimally, after creating the translation script, I would like to have an interactive form where I can "map" fields in my input layer to the output layer (ie- The output layer will have a field called "Address", what is it called in the input layer?)  Some were mentioned in the Q&amp;A at   And here are a couple of related articles that I found. Prove $2^n-1$ is divisible by $(2^m-1)^2$ iff... Prove that for any $m,n\in\mathbb{Z}^+$, the number $2^n-1$ is divisible by $(2^m-1)^2$ iff the number $n$ is divisible by $m(2^m-1)$.  My work so far:  It's an if and only if statement, so the common approach is to split the proof into two "if then" proofs ($A\rightarrow B \text{ and } B\rightarrow A$).  Starting with $A\rightarrow B$, for a direct proof, I would assume $A$ and prove $B$. I can use the definition of divisibility:  Assume $n=m(2^m-1)k$ for some integer $k$  Then I need to prove that $2^n-1=(2^m-1)^2h$ for some integer $h$  So I'm thinking I need to start with the expression $2^n-1$ and transform it to $(2^m-1)^2h$, or vice versa, and I need to use the assumption $n=m(2^m-1)k$ somehow.  I notice that the LS of the equality contains the variable $n$ while the RS contains $m$, and the assumption relates $n$ to $m$, so that could be a way forward. For example, simple substitution gives $2^{m(2^m-1)k}-1$, but I'm not sure what I could do from here. Perhaps there's a divisibility theorem that could help?  Other things I notice: $2^n-1$ and $(2^m-1)^2$ are always odd. Also, the divisors in both cases are multiples of $2^m-1$.  Any help/hints appreciated. If $m_1 , m_2, \cdots m_n$ are natural numbers where at least one of them is not a perfect square, then how do I prove that the sum $$\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$$ is irrational? I'm hoping to prove this using methods from pre-calculus level algebra. What is the preferable setup for MiKTeX to keep all packages up-to-date easily? I need to show that for prime numbers of the form $p=4n+1$, $x=(2n)!$ solves the congruence $x^2 \equiv-1\pmod p$. I then need to show this implies p isn't a gaussian prime.   I have started to solve this using Wilson's theorem that a number $z$ is prime iff $(z-1)!\equiv-1\pmod z$. Therefore the endpoint of my proof should be that $(p-1)!\equiv-1\pmod p$.  As $p$ is of the form $p=4n+1$, I only need to prove that $4n!$ is congruent to $-1$ modulo $p$.   Here is my working so far: Starting with the congruence $x^2 \equiv -1\pmod p$:    $$\eqalign{x^2 = -1\pmod p&amp;\implies   x^2 + 1 = kp \implies (x-i)(x+i) = kp\cr &amp;\implies ((2n)!-i)((2n!)+i))=kp \implies 4n!- 1 =kp\cr}$$   This is where I start to run out of any ideas that seem to get me anywhere. Any tips would be greatly appreciated!