In the book Analysis of Financial Time Series by Rue Tsay, I read:     A time series $\{p_t\}$ is a random walk if it satisfies $p_t = p_{t−1} + a_t$    where $p_0$ is a real number denoting the starting value of the process   and $\{a_t\}$ is a white noise series. If $p_t$ is the log price of a   particular stock at date $t$ , then $p_0$ could be the log price of the   stock $a_t$ its initial public offering (IPO) (i.e., the logged IPO   price). If $a_t$ has a symmetric distribution around zero, then   conditional on $p_{t−1}$, $p_t$ has a 50–50 chance to go up or down, implying   that $p_t$ would go up or down at random. If we treat the random-walk   model as a special AR(1) model, then the coefficient of $p_{t−1}$ is unity,   which does not satisfy the weak stationarity condition of an AR(1)   model. A random-walk series is, therefore, not weakly stationary, and   we call it a unit-root nonstationary time series.   If $p_t$ has a 50–50 chance of going up or down, then its mean is constant, correct? So if $p_0=1$, and it can go to 0 with a probability of 0.5 or go to 2 with probability of 0.5, then the mean is constant at 1.  So why is this not a stationary process?
The  that is defined as $Y_{t} = Y_{t-1} + e_t$, where $e_t$ is white noise. Denotes that the current position is the sum of the previous position + an unpredicted term. You can prove that the mean function $\mu_t = 0 $, since $$E(Y_{t}) = E(e_1+ e_2+ ... +e_t) = E(e_1) + E(e_2) +... +E(e_t) = 0 + 0 + \cdots + 0$$ But why is it that the variance increases linearly with time? Does this have something to do with it's not “pure” random, since the new position is very correlated with the previous one? EDIT: Now I have a much better understanding by visualizing a big sample of random walks, and here we can easily observe that the overall variance does increase over time,  and the mean is as expected around zero. Maybe this was trivial after all, since in the very early stages of the time series (compare time = 10 with time = 100) the random walkers have not had yet the time to explore as much.
Pre-declare all private/local variables? File /etc/ld.so.nohwcap is missing in Debian 7 What is the file /etc/ld.so.nohwcap used for?. In my machine (Debian 7), this file is not found. Therefore, Which package or library this file belongs to? What is this lock screen with ads and how do I remove it? I think my device got a malware. A lock screen with ads sometimes appear, and I must swipe an area at the bottom of the screen to open the phone. I am in Eastern Europe, some ads are in Slavic languages, and they are seemingly related to travel.    Image resized. Click to view the full image  I have seen ads for Lot Air (Polish carrier). I recently went to their site.  Considering my recent surfing activity and the nature of the ads, I guess the malware is connected to airlines active in Eastern Europe.  The lock screen also promotes the app "DU Quick Charger" by claiming it has recharged the phone extra fast during screen off. (!?) It also shows battery level. I have never installed DU Quick Charger. I doubt DU Quick Charger is installed on the phone.  The problem started after installing Ryanair app. The problem is maybe, but not surely, related to this. Uninstalling Ryanair and restarting the phone doesn't remove the occasional appearance of this lock screen. After a while the situation has changed for the worse, now this lock screen appears after EVERY screen off.  I have always used Avast. I am not rooted and only install via Google Play.  What is this? And how do I remove it? sed substitution with Bash variables Can I kill an indestructible creature with -X/-X and sufficient damage marked on it? Can I hit a 4/4 indestructible creature with a  (3 damage) and give it -1/-1 with  to kill it? (Does it matter which way around I cast those?)  I read  here and it says, damage is removed in cleanup step, so it should be on the creature still after both spells.     702.12. Indestructible   702.12a Indestructible is a static ability.   702.12b A permanent with indestructible can’t be destroyed. Such permanents aren’t destroyed by lethal damage, and they ignore the state-based action that checks for lethal damage (see rule 704.5g).   702.12c Multiple instances of indestructible on the same permanent are redundant. Capture stdout and stderr into different variables Trying to get my head around the commutator subgroup. This is an excercise from Artin's Algebra:  Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian.  Here is what I've done:  Let $xC,yC \in G/C$ then $xyx^{-1}y^{-1}C = C$ since $xyx^{-1}y^{-1}$ is a commutator hence belongs to $C$. But then $xyC = yxC$ so $xC$ and $yC$ commute in $G/C$. This can be done for any elements, so $G/C$ is abelian.  This seems somewhat surprisingly short. Is that all there is to it?   Regards Calculate the display width of a string in C#? A Java version of this question was just answered, and, well, I don't know how to do this in .net.   So how do you calculate the display width of a string in C# / .net? I updated LyX to 2.3.0-1, and found that LyX was displaying all the "-" characters in Math Mode as "\lyxminus". No such problem occurs when entering outside of Math Mode. LyX also seems to have no problem compiling to TeX, and the PDF output looks normal.  I removed and re-installed LyX, and this issue was still there.   Does anyone have any ideas on what might be going on? I am using Linux Mint 18.2 Sonya.  Note: I previously saw this question ask about the same thing, but the author appears to have deleted their post: It just struck me that Utility Companies world-wide supply domestic users at either 50/60Hz. It could have been selected to be 10 Hz, or 100 Hz ... whatever  Was 50/60 chosen at random? Is there some rationale involved (efficiency, easier to construct/control etc) in restricting domestic supply to either 50/60 Hz?  In other words, if I were designing a power system, why would I pick 50/60 Hz instead of some other frequency? OLS is BLUE. But what if I don't care about unbiasedness and linearity? How can I show or hide boot messages when Ubuntu starts? Is there a way to easily turn on/off showing the boot messages (loading the services) when Ubuntu starts? Is it something in Grub2?  I am running 10.04. How do I disable client side decoration globally in GNOME? I'm using Ubuntu GNOME 17.04. One encounters client side decoration (CSD) as most of the official GNOME applications (Nautilus, gedit, Evince etc.) use  (a combined title bar and toolbar, see linked screenshots) whereas other applications don't. This feature is disabled in Unity, all the applications use traditional separate title bars.  Screenshots (from ):     Nautilus with GtkHeaderBar   Nautilus without GtkHeaderBar     For consistency I want to disable GtkHeaderBars globally (in all the applications using it) and use separate title bars. Is there a way to achieve that?  I have heard of gtk3-nocsd package, but couldn't make it work with GNOME for all the applications. If $a_0+\frac{a_1}{2}+\dots + \frac{a_n}{n+1}=0$ then $a_nx^n+\dots + a_0$ has a root in $[0,1]$ Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots +a_0$ be a real polynomial such that $a_0+\frac{a_1}{2}+\frac{a_2}{3}+\dots + \frac{a_n}{n+1}=0$.  Show that $P$ has a root in $[0,1]$. I would like proof verification and alternative proofs.  My solution:      notice that the antiderivative of $P(x)$ is $\frac{a_nx^{n+1}}{n+1}+\frac{a_{n-1}x^n}{n}+\dots + \frac{a_1x^2}{2}+a_0x+C$. From here $\int_{0}^1P(x)dx=a_0+\frac{a_1}{2}+\frac{a_2}{3}+\dots + \frac{a_n}{n+1}=0$. If $P$ had no roots then $P$ would be a strictly positive(or negative) continuous function, so its integral would be strictly positive ( or negative), a contradiction. If I'm standing at the equator, jump, and land 1 second later, the  Earth does NOT move 1000mph (or .28 miles per second) relative to me,  since my velocity while jumping is also 1000mph.   However, the Earth is moving in a circle (albeit a very large one),  while I, while jumping, am moving in a straight line.   How much do I move relative to my starting point because of this? I  realize it will be a miniscule amount, and not noticeable in practise,  but I'd be interested in the theoretical answer. $|f(x)-f(y)|\geq k|x-y|$.Then $f$ is bijective and its inverse is continuous. My exercise says: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ a continuous function e suppose that exists $k$ such that:  $$|f(x)-f(y)|\geq k|x-y|$$  Then $f$ is bijective and  its inverse is continuous.  Well, there's a Theorem , Invariance of domain, that says   "If $U$ is an open subset of $\mathbb {R^n}$ and $f : U \rightarrow\mathbb{R}$ is an injective continuous map, then $V = f(U)$ is open and $f$ is a homeomorphism between $U$ and $V$".  But I'm not knowing how to proceed...need a clue...Thanks for attention!!! How to write a query for exact pairs of groups So I had an exam today and one of the questions were: Give an example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points. At first I had no idea how to do it then I came up with this, even though I know it's wrong:  $$f(x)= \begin {cases} \ \sqrt{x}&amp;\text{if }x &lt;0\\ \sqrt{-x}\ &amp;\text{if }x &gt; 0\\ \end {cases}$$  Like I said I know it's wrong..but can someone give me an example of this? Thanks. Complex roots of $z^3 + \bar{z} = 0$ Rasterio + QGIS - rasterio.drivers() -> TypeError: 'module' object is not callable I have been trying to get rasterio working with my QGIS setup so that I can take advantage of the fast raster manipulations.  After a day of installing / uninstalling and then piece by piece searching and getting components installed one by one where needed (From upgrading GDAL, numpy, and a whole slew of other modules which were fighting each other) I manged to get to the point where I could import rasterio without any errors.  However now when I run, the first call in the example causes a Traceback:  with rasterio.drivers(CPL_DEBUG=True):     creates error =>  TypeError: 'module' object is not callable   I have noticed that other rasterio objects seem fine, but all functions that are in .pyc files have this same issue.  Any help?  - I will continue in the morning, but I was hoping some elves would appear and help solve this one tonight! :-)