I have to show the following inequality: $$ \sum_{k=0}^{\infty} P[Y&gt;k] \leq E[Y] \leq \sum_{k=1}^{\infty} P[Y&gt;k]$$ under the condition that Y is a non negative continuous random Variable.  My first idea was to rewrite the inequality using integrals, but then i end up having to integrate over a sum, which is nothing im particulary excited about. Any Ideas on how i can do this?
I'm working through a problem regarding expected values in Markov chains, and at some point it says:     Recall from probability that if $X$ is a positive integer valued random variable, then $\mathbb{E}[X] = 1 + \sum^\infty_{k=1}\mathbb{P}(X &gt; k)$.   I know that by definition $\mathbb{E}[X] = \sum_a a\mathbb{P}(X = a)$, but I can't see how the above equality follows from this, nor am I sure if this is how to approach the problem.
Optimal estimation of a single parameter given data at different scales Let $A,B,C,D$ be independent normal random variables distributed according to $\mathcal{N}(0,\sigma^2)$ and $E = A+B+C+D$.  The goal is to provide an optimal estimate of $\sigma^2$ given samples of $A,B,E$ but not $C,D$.  In this case optimal refers to minimising the root-mean square deviation of the estimate $\hat{\sigma}^2$:  $$RMSD(\hat{\sigma}^2) = \sqrt{\mathbb{E}\left[\left(\hat{\sigma}^2-\sigma^2\right)^2\right]}$$  I can think of a simple approach to estimate $\sigma^2$ given samples of $A,B$ which is to simply compute the sample variance. Also I know that the variance of $E$ is $4\sigma^2$ so I could compute the sample variance of $E$ and divide it by 4 to get an estimate of $\sigma^2$. However it should be possible to combine the information from both $E$ and $A,B$ to improve estimation $\sigma^2$. Could you outline the approach to use both sets of information? Industry standard file format deliverables for logos? Bring all planes in mesh to same level I have added a plane in top view. I have then extruded the plane according to a floorplan in topview.  I don't know what happened, but when I switched my view, I could see that some parts of it are for some reason high up in the air.  I would like to ask how I can bring everything back down onto the grid / floor.  And if somebody could tell me how this might have happened, I would also be glad. We were given the following problem as an assignment, we proved the converse in class for prime numbers; however, we can't use the assumption that $\operatorname{gcd}(a,b) = 1$.  So I am struggling to think of my first possible move.  We are given $ar + bs = 1$, the only think I can think of is let $d$ be a common divisor or $ar + bs$ and then show that the $d = 1$.  However, I am not really sure how to do that.  If anyone could just give me a hint or two to nudge me in the right direction I would be appreciative. How can you dynamically create variables via a while loop? Do you really answer "How do you do?" with "How do you do?" Place label in middle of line with TikZ What is the best way to place a label in the middle of a line with TikZ? I mean smack dab in the middle of the line, not above or below it. The following is close  \begin{tikzpicture} \draw[|-&gt;, color=red] (-4cm, 0.5cm) -- (2.5cm, 0.5cm) node[midway, color=black] {$\tau_{r}$}; \end{tikzpicture}   But the line crosses right through the node and its text, which is bad.  I don't know if there is an option to choose the background color of a node, which I think would solve the problem (e.g. background=white). fill doesn't seem to work. Can I use Ctrl+Shift for changing languages? I can't set this hotkey for changing keyboard languages.  At this moment Alt+Shift switch languages, but I can't see this hotkey in keyboard hotkeys list or in   gsettings list-recursively  org.gnome.desktop.wm.keybindings | sort | more   output. This is problem 6.2 from the 3rd edition of Principles of Mathematical Analysis.      Problem 6.2: Suppose $f\geq 0$, f is continuous on $[a, b]$, and $\int_a^b f(x) \, dx = 0$. Prove that $f(x)=0$ for all $x \in [a, b]$.   I'm looking for a critique of my proof. It's a pretty easy problem, but I am always wary of making too bold of assumptions, especially on these low level/fundamental proofs. I'll be using Rudin's notation and refer to theorems from the text (If I should include the text of each theorem, feel free to leave a comment... I'm lazy but could probably use the TeX practice :p)  Proof: Assume, for contradiction, that $f&gt;0$. Then, for any partition $P$ we have the Lower Riemann Sum: $L(P, f)=\sum_{i=1}^n m_i \, \Delta x_i$. At least one $\Delta x_i$ must be positive, since $a &lt; b$, and each $m_i$ must be positive since we have $f&gt;0$ by assumption, so certainly $\sup f &gt; 0$. That means $L(P, f)&gt;0$. Thus, we have: $$0 = 0(b-a) &lt; L(P,f)\leq \sup L(P, f) = \inf U(P,f) =L$$ where the last string of equalities holds because our function is continuous on a compact interval, so is integrable by theorems 6.8 and 6.6. So our integral has value $L&gt;0$. This is in contradiction to our given assumption that $\int_a^b f(x) \, dx = 0$, so we must have that $f=0$ on $[a, b]$. $\Box$  So, I am wondering if my proof is correct (and is presented well). Also if someone could enlighten me as to what Rudin means when he say "Compare this with exercise 1," I'd be appreciative. Is it a hint or is there something else he expects you to notice? there are a lot of things I could compare :)...  Exercise 6.1: Suppose $\alpha$ increases on $[a, b]$, $a \leq x_0 \leq b$, $\alpha$ is continuous at $x_0$, $f(x_0)=1$, and $f(x)=0$ if $x\neq x_0$. Prove that $f$ is Riemann-Stieltjes Integrable and that $\int f \, d\alpha = 0$ How to convert aspell dictionary to simple list of words? I want to get list of all words from aspell dictionary. I downloaded aspell and aspell polish dictionary, then unziped it using:  preunzip pl.cwl   I got pl.wl:  ... hippie hippies hippiesowski/bXxYc hippika/MNn hippis/NOqsT hippisiara/MnN hippiska/mMN hippisowski/bXxYc ...   but they appear with sufix like /bXxYc or /MNn. These suffixes are defined in pl_affix.dat, which looks like  ... SFX n Y 5 SFX n   a         0         [^ij]a SFX n   ja        yj        [^aeijoóuy]ja SFX n   a         0         [aeijoóuy]ja SFX n   ia        ij        [^drt]ia SFX n   ia        yj        [drt]ia ...   It is connected to the declination and conjugation. How can I add to the first list all forms (with all corresponding suffixes as defined in .dat file ) ?  BTW: I need this list to spell-checker jazzy. As a grad student, I just find it tremendously frustrating when emails are ignored by faculty (both at my institution and at other places). Is this acceptable? I'd be content with a simple response with one line along the lines of "I have read this email and am busy right now, will respond in X days when I can write a more detailed email." I understand that, as a first year student in my field, I'm not really in a position to request anything of anyone, but it just seems like common decency.  Instead, I never hear back, as if my email has been jettisoned off into oblivion. regular expressions / pattern matching on ubuntu command line Making images from single.php pointing to an attachment .php template Why is the mol a fundamental physical quantity? I am starting to study physics in detail and as I read about physical quantities, I was puzzled why mol (amount of substance) is taken as a physical quantity.  A physical quantity is any quantity which we can measure and has a unit associated with it. But a mol represents the amount of substance by telling the number of particles (atoms, molecules, ions, etc.) present. So it is a pure number and numbers are dimensionless. So mol should not be considered a physical quantity.  Also, fundamental physical quantities should be independent of each other. I am wondering whether mass and mol are independent. This is so as they surely affect each other as we can evidently see while calculating the number of moles and using the mass of that sample for calculation.  So how is the mol a fundamental physical quantity and independent of mass? $f(z)$ and $\overline{f(\overline{z})}$ simultaneously holomorphic Prove that the functions $f(z)$ and $\overline{f(\overline{z})}$ are simultaneously holomorphic.   I take this to mean that $f(z)$ is holomorphic if and only if $\overline{f(\overline{z})}$ is holomorphic.  Let $g(z)=\overline{f(\overline{z})}$. Note that $\overline{g(\overline{z})}=f(z)$. So it suffices to prove that if $f(z)$ is holomorphic, then $g(z)$ is holomorphic.  Write $f(z)=u(z)+iv(z)$. Since $f(z)$ is holomorphic, the real and imaginary parts satisfy the Cauchy-Riemann equations:  $$\frac{\partial{u(z)}}{\partial{x}} = \frac{\partial{v(z)}}{\partial{y}}, \frac{\partial{u(z)}}{\partial{y}} = -\frac{\partial{v(z)}}{\partial{x}}.$$  We have $g(z) = u(\overline{z})+i(-v(\overline{z}))$. To prove that $g(z)$ is holomorphic, we must prove that its real and imaginary parts satisfy the Cauchy-Riemann equations:  $$\frac{\partial{u(\overline{z})}}{\partial{x}} = \frac{\partial{(-v(\overline{z}))}}{\partial{y}}, \frac{\partial{u(\overline{z})}}{\partial{y}} = -\frac{\partial{(-v(\overline{z}))}}{\partial{x}}.$$  How can we obtain this from the above relations? How to select the most recent set of dated records from a mysql table How to control object's path follow speed based on its position What is the difference between “do” and “make”? How and when is old-releases.ubuntu.com updated? Rename a file to parent directory's name in terminal I have to deal with a large number of files nested inside of directories (one file per directory), resembling something like this:  fred/result.txt john/result.txt mary/result.txt ...   I am currently using the following command to process each of those files:  find . -maxdepth 1 -type d \( ! -name . \) -exec bash -c \"cd '{}' &amp;&amp; processResult result.txt\" \;   I am looking for something I can add into the end of this command that will change the filenames to fred.txt, etc. and then later move the file into the parent directory to eliminate the extra layer of directories.  What would be the best way to do this?