I started reading the book Brownian motion by Mörters and Peres where I don't quite understand the following example. For completeness I've provided the definitions for brownian motion and finite dimensional distributions used in the book.  Definition $1.1$$ . \quad$ A real-valued stochastic process $\{B(t): t \geqslant 0\}$ is called a (linear) Brownian motion with start in $x \in \mathbb{R}$ if the following holds: $\bullet\hspace{0,2cm} B(0)=x,$ $\bullet\hspace{0,2cm}$the process has independent increments, i.e. for all times  $0 \leqslant t_{1} \leqslant t_{2} \leqslant \ldots \leqslant t_{n}$ the increments $B\left(t_{n}\right)-B\left(t_{n-1}\right), B\left(t_{n-1}\right)-B\left(t_{n-2}\right), \ldots, B\left(t_{2}\right)-B\left(t_{1}\right)$ are independent random variables, $\bullet\hspace{0,2cm}$ for all $t \geqslant 0$ and $h&gt;0,$ the increments $B(t+h)-B(t)$ are normally distributed with expectation zero and variance $h$ $\bullet\hspace{0,2cm}$ almost surely, the function $t \mapsto B(t)$ is continuous. We say that $\{B(t): t \geqslant 0\}$ is a standard Brownian motion if $x=0$. By the finite-dimensional distributions of a stochastic process $\{B(t): t \geqslant 0\}$ we mean the laws of all the finite dimensional random vectors $$ \left(B\left(t_{1}\right), B\left(t_{2}\right), \ldots, B\left(t_{n}\right)\right), \text { for all } 0 \leqslant t_{1} \leqslant t_{2} \leqslant \ldots \leqslant t_{n} $$ To describe these joint laws it suffices to describe the joint law of $B(0)$ and the increments $$ \left(B\left(t_{1}\right)-B(0), B\left(t_{2}\right)-B\left(t_{1}\right), \ldots, B\left(t_{n}\right)-B\left(t_{n-1}\right)\right), \text { for all } 0 \leqslant t_{1} \leqslant t_{2} \leqslant \ldots \leqslant t_{n} $$    Example 1.2 Suppose that $\{B(t): t \geqslant 0\}$ is a Brownian motion and $U$ is an independent random variable, which is uniformly distributed on [0,1] . Then the process $\{\tilde{B}(t): t \geqslant 0\}$ defined by $$ \tilde{B}(t)=\left\{\begin{array}{ll} B(t) &amp; \text { if } t \neq U \\ 0 &amp; \text { if } t=U \end{array}\right. $$ has the same finite-dimensional distributions as a Brownian motion, but is discontinuous if $B(U) \neq 0,$ i.e. with probability one, and hence this process is not a Brownian motion.  Now my questions are  Why does $\tilde{B}(t)$ have the same finite-dimensional distributions as a Brownian motion ? Why is $B(U)\neq 1$ with probability one ?
In the book on Brownian Motion by Yuval Peres, he makes the following claim:     Suppose that $\{ B(t); \ t \geq 0 \}$ is a Brownian Motion and $U$ is an independent random variable, uniformly distributed on $[0, 1]$. Then the process $\{ \tilde{B} (t); \ t \geq 0 \}$ defined as,   $$\tilde{B}(t) = \begin{cases} B(t)  &amp; \mbox{ if $t \neq U$} \\ 0  &amp; \mbox{ if $t = U$} \end{cases}$$      has the same finite-dimensional distributions as a Brownian motion, but is discontinuous if $B(U) \neq 0$, i.e. with probability one, and hence this process is not a Brownian motion.   I'm having difficulty in proving that the finite dimensional joint distribution ( basically the joint distribution of $( \ \tilde{B}(t_1), \ \tilde{B}(t_2), \tilde{B}(t_3), \dots \tilde{B}(t_n) \ )$ ) is the same as the finite dimensional distribution of $B(t)$. Can you please give me some clues.
Suppose $Y_1, \ldots, Y_n$ is a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. Let $S^2$ be the sample variance, which is unbiased for $\sigma^2$.  GOAL: Show that $S = \sqrt{S^2}$ is a biased estimator of $\sigma$.  I realize that $(n-1)S^2/ \sigma ^2$ is chi-square with $(n-1)$ degrees of freedom.  I am given that $$E(S) = E\left[\frac{\sigma}{\sqrt{n-1}} \left[\frac{(n-1)S^2}{\sigma^2}\right]^{1/2}\right] = \frac{\sigma}{\sqrt{n-1}} \int_0^\infty v^{1/2} \frac{1}{\Gamma[(n-1)/2] 2^{(n-1)/2}} v^{(n-1)/2} e^{-v/2} \, dv$$  Can anyone help me understand why they have $v^{1/2}$ and $v^{(n-1)/2}$ (combining them $v^{n/2}$ as I think it should be $v^{1/2}$ and $v^{\frac{(n-1)}{2}-1}$ (combining both terms give $v^{(n-2)/2}$ since the pdf for chi-square distribution is $$ f(v) = \frac{v^{(x/2)-1} e^{-v/2}}{2^{x/2}\Gamma(x/2)},$$ where $x$ is degree of freedom?  Could anyone please tell me which one is the correct answer with explanation?  Thank you How to zero out a drive? I'm currently running Ubuntu 12.04 and want to completely format my laptop so I can install Windows 7 (replace Ubuntu).   When I put in my bootable USB with the OS it just shows a black screen with a white blinking cursor on the top left corner. I wait for hours (to be specific 5 whole hours) waiting for something to happen but I get nothing, so I pull out the USB and my Ubuntu 12.04 loads up.  Repeated this several times with putting different boot priority options on top-est that relate to USB but the results are same. I go and visit some sites on the net like this  that say I have to zero out my hard drive.   My question is how?  Note: Erasing out all my data is no problem cause I have nothing important to backup.  If I have to lose my primary OS (Ubuntu 12.04) in the process I am ready to as my aim is just to install Windows 7 successfully. Please don't answer that there is something wrong with my USB or my USB reader/port/hardware or the content inside the USB cause they all works fine on my brothers PC as it boots up flawlessly. Obtaining a formula for prediction limits in a linear model (i.e.: prediction intervals) Let's take the following example:  set.seed(342) x1 &lt;- runif(100) x2 &lt;- runif(100) y &lt;- x1+x2 + 2*x1*x2 + rnorm(100) fit &lt;- lm(y~x1*x2)   This creates a model of y based on x1 and x2, using a OLS regression.  If we wish to predict y for a given x_vec we could simply use the formula we get from the summary(fit).  However, what if we want to predict the lower and upper predictions of y? (for a given confidence level).  How then would we build the formula? Is "You are correct" correct? It seems to me that "You are correct" means that the person in question is supposedly "correct" in their quiddity. "That is correct" or "You are right" seems a lot better.  I also would like to know if someone is using correct(ly) correctly, as to correct someone who is potentially using correctly incorrectly. System.FinalException: Record is read-only: Trigger.updateCompetitors: line 24 How is an instance initializer different from a constructor? Background I updated my TeX Live installation on my Mac a while ago and got the error LaTeX3 Error: Mismatched LaTeX support files detected. I found a solution at .  As far as I understand it (which isn't very far), there are system formats and local formats, and TeX Live is updating the wrong one.  The fix is to update the formats. Using fmtutil-user --all worked for me.  It spewed out a load of stuff, then output fmtutil [INFO]: log file copied to: /Users/&lt;user&gt;/Library/texlive/2020/texmf-var/web2c/luatex/dvilualatex.log fmtutil [INFO]: /Users/&lt;user&gt;/Library/texlive/2020/texmf-var/web2c/luatex/dvilualatex.fmt installed.  I don't know what this tells me, but my TeX Live preferences has TeX programs set at /Library/TeX/texbin, which seems to be a link to /usr/local/texlive/2020/bin. Problem Every time I update TeX Live, I have to apply the solution again. Question Is there a way to make TeX Live use the correct formats permanently so that I don't have to keep running fmtutil-user --all after every TeX Live update?  Extra information  to  check if you have some local formats around which hide the system formats. (with eg kpsewhich --all --engine=pdftex pdflatex.fmt)  When I did that, it output /Users/&lt;user&gt;/Library/texlive/2020/texmf-var/web2c/pdftex/pdflatex.fmt /usr/local/texlive/2020/texmf-var/web2c/pdftex/pdflatex.fmt  but I don't know what that's telling me. What are your tips for improving overall system performance on ubuntu? Inspired by this  I realized that some default settings may be rather conservative on Ubuntu and that it's possible to tweak it with little or no risk if you wish to make it faster.  This is not meant to be application specific (e.g. make firefox load pages faster), but system wide.  Preferably 1 tip per answer, with enough detail for people to implement it.  A couple of mine would be:   Install  (via Software Center or sudo apt-get install preload); Change  value - "which controls the degree to which the kernel prefers to swap when it tries to free memory";   What are yours?  PS: Since this is not intended to have a unique answer but rather, several useful tips, I'm making this community wiki out-of-the-box. Because of past conflicts, I don't believe my advisor will write me a good LOR for postdoc positions. What should I do? How do candles and wicks work? I have two exchange accounts on my iPhone 4. One that connects to my gmail, the other connects to my work's Lotus Notes. There are no overlapping contacts in the two accounts. There are also no contacts saved locally on my phone.   When I enable contacts in both accounts, I get two identical entries for each of the contacts in my gmail. If I disable the Notes exchange account, then there are no more duplicates. If I have Notes contacts enabled, and gmail contacts disabled then I have no contacts at all.  Does anyone know what is going on? Is this a problem with having multiple exchange accounts or is it a Lotus Traveler problem? (I tend to blame Notes for all the trouble in the world, but that is just my bias)  Thanks all. How do I make ditto marks? In France, students are taught not to repeat the same text twice when there is little difference between a line and the one below. It's a quick way to take notes.  I reproduced this effect by tinkering:    The code is as follows:  \documentclass[aspectratio=1610]{beamer}  \usepackage[T1]{fontenc} \usepackage[french]{babel}  \usepackage{amsmath, amsfonts, amssymb} \usefonttheme[onlymath]{serif}  \begin{document} \begin{frame} Lorsque le \og \textit{carreau carré} \fg{}   mesure: \begin{itemize} \item $\rm 1\,cm$ de côté, l'unité d'aire est appelée le \og \textrm{centimètre carré} \fg{} et est noté $\rm cm^2$; \item $\rm 1\,dm$ \hspace{1mm}'' \hspace{3mm}'' \hspace{8mm}'' \hspace{8mm}'' \hspace{5mm}''\hspace{8mm}'' \hspace{5.5mm}''\hspace{1mm} \og \textrm{décimètre carré} \fg{} \hspace{2mm}'' \hspace{2mm}'' \hspace{4mm}'' \hspace{1.5mm} $\rm dm^2$; \item $\rm 1\,m$ \hspace{3mm}'' \hspace{3mm}'' \hspace{8mm}'' \hspace{8mm}'' \hspace{5mm}'' \hspace{7mm}'' \hspace{5.5mm}''\hspace{1mm} \og \textrm{mètre carré} \fg{}  \hspace{9mm}'' \hspace{2mm}'' \hspace{4mm}'' \hspace{1mm} $\rm m^2$; \item etc. \end{itemize} \end{frame} \end{document}   How can I get the same effect more simply than by tinkering like I just did? I need to find the values of $a$ and $n$ in the following  $$ (1 + ax)^n  = \sum_{i=0}^\infty \binom{2i}{i}x^i$$  How can I compare things and show that $a=-4$ and $n = -{1 \over 2}$. It's easy to expand $1 \over \sqrt{1 -4x}$ see that it's valid but I'm looking other way around. Is there a value that can be assigned to my time, i.e. an hourly rate, so that I can be given a receipt to remit to CRA come Income Tax Filing time?  I live in Canada. How to prove floor identities? Mountain Bike on Paved Roads I'm an occasional biker. I bought a mountain bike (nothing fancy) with the idea that I might use it off-road or on mountain trails (of which there are many here in Calgary). However, since I've bought it I've done nothing but bike in the city on streets and paved bike trails.  Is there anything worth doing to my bike to convert it to something more suited to urban biking? In particular, the tires have me wondering; they have large rubber off-roading lugs that surely aren't doing me any favors on pavement. I'm not looking to spend a tons of money on a retrofit, but will do so if its worthwhile. I have a repeated composition of functions ${T_n}(z) = {\tau _0} \circ {\tau _1} \circ {\tau _2} \circ  \cdots  \circ {\tau _n}(z)$  By analogy with $\sum\limits_{i = 1}^n {} ,\prod\limits_{i = 1}^n {} ,\bigcup\limits_{i = 1}^n {} ,\bigcap\limits_{i = 1}^n {} ,$ I want to write ${T_n}(z) = \left( {\mathop  \circ \limits_{i = 0}^n {\tau _i}} \right)(z)$ or even ${T_n}(z) = {\mathop  \circ \limits_{i = 0}^n {\tau _i}} (z)$. Can I do this? Print  only? How do I print the indicated div (without manually disabling all other content on the page)?  I want to avoid a new preview dialog, so creating a new window with this content is not useful.  The page contains a couple of tables, one of them contains the div I want to print - the table is styled with visual styles for the web, that should not show in print. How to train students to write high-quality research papers? As someone supervising postgrad students, I wonder what is the best way to train these students to write high-quality research papers?    The best way seems to be through coauthorship but ...  On the one hand I can be hands-off and rather give them feedback as best as I can but let them do most of the work themselves, sort of like a "sink or swim" strategy.  On the other hand I can be hands-on and rewrite the paper (or large parts of it) and hope that they learn from the example I give.  On the one hand I want them to learn to write papers without me some day and I want them to learn as quickly as possible to take load off me.  On the other hand I want us to publish in the best place possible with as little stress as possible.  On the one hand I want them to see the peer review comments from what they themselves have written.  On the other hand I do not want them to be discouraged by the peer review system early on and do not want to submit papers I know we could have written better.  On the one hand I want them to get confidence by being published and presenting work at good venues.  On the other hand I wonder if they need to get there themselves to get that confidence boost.    At the moment I'm very hands on. There's a few factors for this: I don't like submitting papers that are not up to a certain standard, I'm a native English speaker in a non-English speaking country with non-native speaking students, publications are a high priority for me, etc.  But recently a friend/colleague warned me that I should not be so hands-on with the writing of the papers of my students. He pointed to recently graduated researchers in my area that published good work as students but who struggled after leaving their PhD supervisor saying their supervisor did them no favours in the long term ... that by being so hands on, while the students have success getting published and so forth with me, I'm not really doing the most important thing a good supervisor should do: making myself obsolete.    My question is how to train students to write good research papers and in what ways can I balance being hands-on and hands-off?  I'm also interested in methods not directly involving coauthorship. Graphing $y = \frac{x^2-4}{x-2}$ I'm trying to graph the function $y = \frac{x^2-4}{x-2}$.  I factored the numerator into $(x+2)(x-2)$, and then canceled the common factor (x-2), leaving the function as $y = x + 2$.    Is it acceptable to graph the original function as $y = x + 2$, a straight line with domain and range all real numbers?  Or do you have to restrict the domain as in the original function?  Thanks, Sean