I am an Economics undergraduate who was reading through a textbook on statistical theory. On one of the questions, I had to find the Variance of $X$ the joint probability distribution, $f(x,y)=\frac{1}{4^{x+y}}$, where $x$ and $y$ were discrete random variables $x=0,1,2,...$ and $y=0,1,2,...$ When calculating $Var(x)$, and trying to find $E(x^2)$ I got stuck at the summation for $\frac{X^2}{4^X}$ for $0\le X$. Previously in the part when I calculated $E(x)$, I was able to sum $\frac{X}{4^X}$ using an AGP. However, when looking at the variance portion, I'm not sure what kind of series this is, and what method I can use to derive an answer. Any help would be greatly appreciated !!
I'd like to find out why  \begin{align} \sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6 \end{align}  I tried to rewrite it into a geometric series  \begin{align} \sum_{n=0}^{\infty} \frac{n^2}{2^n} = \sum_{n=0}^{\infty} \Big(\frac{1}{2}\Big)^nn^2 \end{align}  But I don't know what to do with the $n^2$.
Show/Hide  dropdown, with jQuery, based on value Relation bewteen Hessian Matrix and Curvature According to ,  It describes the local curvature of a function.   AFAIK, for one-variable function $f(x)$, its local curvature is  $$\kappa = \frac{|f''|}{(1 + f'^2)^{3/2}},$$ and its Hessian matrix is  $$\mathcal{Hess}(f) = [f''],$$ right? And here is my problem, I think the local curvature is not just described by its Hessian matrix, because $f'$ also has its role in it, doesn't it?  And furthermore, for 2-variable function $f(x,y)$, its Hessian matrix is  $$\mathcal{Hess}(f) = \left[ \begin{array}{cc} f_{xx}'' &amp; f_{xy}'' \\ f_{xy}'' &amp; f_{yy}'' \end{array} \right].$$ How does it relate to the local curvature of $f(x,y)$? Google Voice Recognition on iOS How do I show that there exists variational/action principle for a given classical system? We see variational principles coming into play in different places such as Classical Mechanics (Hamilton's principle which gives rise to the Euler-Lagrange equations), Optics (in the form of Fermat's principle) and even General Relativity (we get Einstein's equation from the Einstein-Hilbert action). However, how do we explain this very principle, i.e., more mathematically, I want to ask the following:      If I am given a set of generalized positions and velocities, say, $\{q_{i}, \dot{q}_{i}\}$, which describes a classical system with known dynamics (equations of motion), then, how do I rigorously show that there always exists an action functional $A$, where    $$A ~=~ \int L(q_{i}, \dot{q}_{i})dt,$$    such that $\delta A = 0$ gives the correct equations of motion and trajectory of the system?    I presume historically, the motivation came from Optics: i.e., light rays travel along a path where $S = \int_{A}^{B} n ds$ is minimized (or at least stationary). (Here, $ds$ is the differential element along the path). I don't mind some symplectic geometry talk if that is needed at all. Fingerprint activation in Dell Vostro 14 in Ubuntu 18.04 I have Dell Vostro 14 with OS Ubuntu 18.04.  I tried   sudo apt install -y fprintd libpam-fprintd   but not able to use fingerprint reader of my laptop. How to escape braces (curly brackets) in a format string in .NET After swapping the positions of the hour and the minute hand, when will a clock still give a valid time? At 12 o'clock, the hour hand and minute hand of the clock can be swapped, and the clock still gives the same time, but at 6 o'clock, it can not be swapped. So in what cases when we swap the hour and the minute hand position does a clock still give a valid time? Can linear momentum be conserved before and after collision in the presence of an external force? I'm a physics tutor for 11th and 12th grade students. I taught my students that momentum is conserved only when the net external force on the system is zero. But in the books we are following, momentum was conserved in the following cases in spite of presence of an external force:   A body (mass m1) is dropped from a tower (height h) and another body (mass m2) is projected vertically upward with initial velocity u. Find the velocities just after collision. (Gravitational force is the external force) Two bodies projected towards each other on a rough table surface, with all required parameters given asked to find the velocities after collision. (frictional force is the external force) Body explodes in mid air. Momentum is conserved before and after explosion. (Again gravitational force is the external force)   I explained to my students in all these cases though an external force is present it has negligible effect. But I'm not satisfied with my answer.  EDIT: Thank you for the comments and answer. It is clear that contribution of external force to change in momentum is negligible. Can someone do the calculation of taking change in momentum into consideration and showing that it is negligible? What's your preferred way of getting current system status (current CPU, RAM, free disk space, etc.) in Python? Bonus points for *nix and Windows platforms.  There seems to be a few possible ways of extracting that from my search:   Using a library such as  (that currently seems not actively developed and not supported on multiple platform) or something like  (again no activity since 2007 it seems and no support for Windows). Using platform specific code such as using a os.popen("ps") or similar for the *nix systems and MEMORYSTATUS in ctypes.windll.kernel32 (see ) for the Windows platform. One could put a Python class together with all those code snippets.   It's not that those methods are bad but is there already a well-supported, multi-platform way of doing the same thing? 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