I'm learning about strongly stationary processes with the definition:  This implies that they have the same distribution for every $t$, I don't really understand what more &quot;same joint distribution&quot; adds. Could someone give a simple example where a process has the same distribution for every $t$ but not the same joint distribution for some indices?
I read that this is because the mean and variance completely specify a normal distribution. However, e.g. for a Poisson distribution the mean, variance and third moment can all be expressed with the same parameter lambda. Is this not what it means to 'completely specify' a distribution?  What is an example of two Poisson distributions having the same mean, variance and different higher moments? If we treated weak stationary as if it was strict stationarity, what would happen when we try to do time series analysis?
Discovering new features $\{a_n\}$ be a sequence such that $ a_{n+1}^2-2a_na_{n+1}-a_n=0$, then $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in... Let $\{a_n\}$ be a sequence of positive real numbers such that   $a_1 =1,\ \  a_{n+1}^2-2a_na_{n+1}-a_n=0, \ \ \forall n\geq 1$.   Then the sum of the series $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in...  (A) $(1,2]$, (B) $(2,3]$, (C) $(3,4]$, (D)$(4,5]$.   Solution attempt:   Firstly, we figure out what $\frac{a_{n+1}}{a_n}$ is going to look like. We get, from the recursive formula, $\frac{a_{n+1}}{a_n}=1+\sqrt{1+\frac{1}{a_n^2}}$ (remembering the fact that $a_n&gt;0$, the other root is rejected).   We know that, if $\lim_{n \to \infty}\frac{a_{n+1}}{a_n}&gt;1$, then $\lim a_n \to \infty$.   Further, $(a_{n+1}-a_n)= \sqrt{a_n(a_n+1)}&gt;0$. (Again, the other root is rejected due to the same reason).   Hence, $(a_n)$ increases monotonically.   Therefore, the largest value of $\frac{a_{n+1}}{a_n}$ is approximately $1+\sqrt{1+\frac{1}{1}} \approx 2.15$  Now, the sum can be approximated as $\displaystyle\frac{\frac{1}{3}}{1-\frac{2.15}{3}} \approx 1.3$ (In actuality, $\mathbb{sum}&lt; 1.3$).  So, option $(A)$ is the correct answer.   Is the procedure correct?   I have been noticing a handful of this type of questions (based on approximations) lately, and the goal is to find out where the sum / the limit of the sequence might lie.   Is there any "definitive" approach that exploits the recursive formula and gives us the value, or does the approach varies from problem to problem? Socket Tester all indicators light up I want to implement designed PID controller. But I am facing the problem of how to limit the saturation limit in both positive and negative direction? I tried using zener diode, but I would like to know if are there any design procedures to get anti-wind up scheme in the  implementation of PID controller using zener/diode combination?  EDIT:  this is the designed pid controller and i am using TL084 op-amps which has +15 volts and -15 volts as their supply. when the error is generated, due to integrator the control signal(Vc) output is going to saturation point of nearly 15 volts. I want Vc to be in the range of (0.85-3.8 volts), so that i can give this to SG3524 PWM IC to generate constant duty ratio PWM pulses. I tried by putting zener diode of 3.3 volt rating, but now Vc is coming upto 3.5 volts(basically saturation limit has come down to 3.5 volts) . The problem is how to limt the Vc to the specified range. Can any one please suggest modifications for proper design of anti-windup  working scheme of this.  Here VFb=-3 volts and VRef= 3 volts How do you configure an OpenFileDialog to select folders? In VS .NET, when you are selecting a folder for a project, a dialog that looks like an OpenFileDialog or SaveFileDialog is displayed, but is set up to accept only folders.  Ever since I've seen this I've wanted to know how it's done.  I am aware of the FolderBrowserDialog, but I've never really liked that dialog.  It starts too small and doesn't let me take advantage of being able to type a path.  I'm almost certain by now there's not a way to do this from .NET, but I'm just as curious how you do it from unmanaged code as well.  Short of completely reimplementing the dialog from scratch, how do you modify the dialog to have this behavior?  I'd also like to restate that I am aware of the FolderBrowserDialog but sometimes I don't like to use it, in addition to being genuinely curious how to configure a dialog in this manner.  Telling me to just use the FolderBrowserDialog helps me maintain a consistent UI experience but doesn't satisfy my curiosity so it won't count as an answer.  It's not a Vista-specific thing either; I've been seeing this dialog since VS .NET 2003, so it is doable in Win2k and WinXP.  This is less of a "I want to know the proper way to do this" question and more of a "I have been curious about this since I first wanted to do it in VS 2003" question.  I understand that Vista's file dialog has an option to do this, but it's been working in XP so I know they did something to get it to work.  Vista-specific answers are not answers, because Vista doesn't exist in the question context.  Update: I'm accepting Scott Wisniewski's answer because it comes with a working sample, but I think Serge deserves credit for pointing to the dialog customization (which is admittedly nasty from .NET but it does work) and Mark Ransom for figuring out that MS probably rolled a custom dialog for this task. In many machine learning applications, the so called data augmentation methods have allowed building better models. For example, assume a training set of $100$ images of cats and dogs. By rotating, mirroring, adjusting contrast, etc. it is possible to generate additional images from the original ones.  In the case of images, the data augmentation is relatively straightforward. However, suppose (for example) that one has a training set of $100$ samples and few hundred continuous variables that represent different things. The data augmentation does not anymore seem so intuitive. What could be done in such case? Add new item in existing array in c#.net How to add new item in existing string array in C#.net?  I need to preserve the existing data. How do I handle the window close event (user clicking the 'X' button) in a Python Tkinter program? How to resolve merge conflicts in Git repository? Prove $0! = 1$ from first principles How can I prove  from first principles that $0!$ is equal to $1$? Possible Duplicate:          When I add a piece of code of figure insertion to the context I'm faced with a problem: a number of irrelevant words appear.    Code:  \begin{figure}     \centering         \includegraphics[width=0.9\textwidth]{C:/Thesis/Latex/thesis_1(1)/Pictures/study area.jpg}         \rule{35em}{0.3pt}     \caption{The Grand St. Bernard wireless sensor network deployment (a) the coordinates of nodes according to the Swiss coordinate system (b) the distribution of the nodes in the study site \citep{r33}}     \label{fig:study area} \end{figure}   Problem: Show $\text{Hom}_{Ab}(A,\mathbb{Q}) = \{0\}$ for finite $A$ Proving that $\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots + \frac{1}{n(n+1)} = \frac{n}{n+1}$ How would we go about proving that $$\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots +\frac{1}{n(n+1)} = \frac{n}{n+1}$$ How do the Flashes breathe utilizing their super speed? The air pressure would be extreme. Do they run in a pressurized bubble? macOS Mojave won't install updates past 10.14.1 due to requirement for APFS formatting I have a MBP 2012 I upgraded from High Sierra to Mojave after it came out as GA. I don't recall if I was asked about upgrading to APFS when Id id that, but at this time the main volume is Mac OS Extended (Journaled).  My current About This Mac...version is Mojave 10.14.1 (18B75). Now, when updates are advertised from the App Store, I'm unable to upgrade because 10.14.3 is required.  No OS updates show in the App Store, and when I searched for the updaters (10.14.2 Combo, etc), I can't install because the installer reports "macOS 10.14.2 Update can't be installed on this disk. This Mac can only install macOS on APFS-formatted drives."  I tried the search results recommending to boot in Recovery Mode and use Disk Utility, but APFS is not available as a type.  I have good backups and I can reformat the boot volume if I need to, but I have not done this before. Should I get a 10.14.3 installer to do this and boot from an external drive? The only guies and message boards I've found just keep saying to install 10.14.3 (I can't) or convert the disk in Recovery Mode (APFS is not available as an option.)  Any known-good ways to get my 10.14.1 upgraded so I can keep getting App Store updates? I am trying to understand the derivation of the stars and bars formula. I understand how a problem can be converted to the stars and bars format, but I am confused about how the problem of choosing all the different ways to put k stars in n sections can be written as ${n+k-1 \choose k}$. Generally "n choose k" is for counting the number of combinations of k objects we can get given n distinct objects right? That I understand since we're doing permutation divided by the number of orderings for each group since order doesn't matter. I need help figuring out how stars and bars fits into this. Adjectives and nouns to describe an answer that doesn't answer the question? I've heard of the adjectives 'nonresponsive' or , but are they restricted to law^?     Collins Dictionary:  =       not responsive, not reacting to a stimulus   The definition above differs from the definitions of adjectives like 'equivocate,  evade' and nouns like , but these are negative and accusive; I prefer neutral connotation.  ^Only if you wish, please see: ,  by David Berg,  by Thomas A. Mauet. Is there a way to access Windows file with administrative rights using WSL?  sudo vi /mnt/c/Users/a/admin/file.txt After looking into the different OAuth types and reading , I'm still confused on the specific pros and cons of user-agent vs web server flows.  The best explanation I can find says that user-agent tends to be used for mobile but it seems that they can both be used for desktop.   What makes it more appropriate to use one over the other in certain scenarios? Can you clone a larger drive onto a smaller drive (used space fits on the smaller) I have a 500GB drive which is about 50GB full. I got a 80GB SSD, which has 74.5GB of available space. I want to clone the system drive onto the SSD.   DriveImage XML will not let me copy a larger drive to a smaller one.