Let $(B_t)_{t \geq 0}$ be standard Brownian Motion with $B_0 = 0.$ Then $$ \mathbb E[\tau_{(a,b)}] = 0 $$ for all $a&lt;b\in \mathbb R$ where $\tau_{a,b}$ denotes the first exit time of $(B_t)_t$, that is $\tau_{(a,b)} = \inf\{ t\geq 0:B_t \notin(a,b) \}$. How do I prove this? Ideas: I know that $B_t - B_s$ is normally distributed for all $s&lt;t$  with mean $0$ and variance $t-s$. Thus, you could maybe use this information to prove the statement above.
I am trying to solve the following:  Let $W_t$ be a brownian motion and $a,b &gt;0.$  Define $\tau$ to be the exit time of $W_t$ from $[-a,b]$, that is $$\tau = \inf\{t \ge 0\ :\ W_{t}(\omega) \notin [-a,b]\}.$$  Show that $\tau$ is integrable and compute $\mathbb{E}(\tau)$.  I have seen this stated without proof multiple places that $\mathbb{E}(\tau)=ab.$  What is the logical argument that backs up this statement?  How are we guaranteed $\tau$ is integrable?
How do I install drivers for Realtek RTL8812BU? So I've been experimenting with Ubuntu on and off for a while, but I honestly have no idea what the hell I'm doing. So I have a realtek RTL8812bu WiFi dongle that I'm hoping to get working. I did some research, and came across  that a couple people across two threads said worked for them:  At the bottom it said there is a new version . So somewhere it said it was confirmed to work on kernels up to 4.16, so I checked and I'm on 4.15.0. But the problem I'm having, is what am I supposed to do? There is some installation guide at the bottom that is about as clear as mud to me since I don't know much. Prove $[\sin x]' = \cos x$ without using $\lim\limits_{x\to 0}\frac{\sin x}{x} = 1$ I came across this question:   From the comments, Joren said:     L'Hopital Rule is easiest: $\displaystyle\lim_{x\to 0}\sin x = 0$ and $\displaystyle\lim_{x\to 0} = 0$, so $\displaystyle\lim_{x\to 0}\frac{\sin x}{x} = \lim_{x\to 0}\frac{\cos x}{1} = 1$.   Which Ilya readly answered:     I'm extremely curious how will you prove then that $[\sin x]' = \cos x$   My question:  is there a way of proving that $[\sin x]' = \cos x$ without using the limit $\displaystyle\lim_{x\to 0}\frac{\sin x}{x} = 1$. Also, without using anything else $E$ such that, the proof of $E$ uses the limit or $[\sin x]' = \cos x$.    All I want is to be able to use L'Hopital in $\displaystyle\lim_{x\to 0}\frac{\sin x}{x}$. And for this, $[\sin x]'$ has to be evaluated first.    Alright... the definition that some requested.   Def of sine and cosine: Have a unit circumference in the center of cartesian coordinates. Take a dot that belongs to the circumference. Your dot is $(x, y)$. It relates to the angle this way: $(\cos\theta, \sin\theta)$, such that if $\theta = 0$ then your dot is $(1, 0)$.  Basically, its a geometrical one. Feel free to use trigonometric identities as you want. They are all provable from geometry. Why do heavier objects fall faster on hills but not in straight drops? All other things being equal, if a heavier object will roll at a higher speed down hill than a lighter one, because it's heavier with more mass, then how is it that dropped objects fall at the same speed regardless of their weight? Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$ Area 51 login no longer redirects nor logs in Is it possible to mark or modify a player with a specific item in their inventory I'm building a CTF map of sorts, using command blocks in vanilla 1.8. I've discovered two commands for detecting if an item is in a players inventory:  /clear @a[team=Red] banner 4 0  /testfor @a[team=Red] {Inventory:[{id:"minecraft:banner"}]}  Those work fine, unless you want to perform a command ON the player carrying that item. For example: I want to cause the player who picks up a banner of the opposite team, to automatically equip that banner in their head slot with a command similar to this:  /replaceitem entity @a[???????] slot.armor.head banner 1 4  Can this be done? If not, is there a way to mark a specific player carrying an item using scoreboards? Word for people who live in the same city What are the people who live in the same city are called? Any words for that? I want to use it in the following context:      I and my ____ are happy. How can I prove formally that the projective plane is a Hausdorff space? Why doesn't the height of a container element increase if it contains floated elements? java.lang.LinkageError: javax.servlet.jsp.JspApplicationContext.getExpressionFactory Restore console session after remote desktop disconnects? Are mounts proficient in armour (barding)? Finding centrelines from polygons in QGIS? I was wondering if there is a way to create centre lines from polygons in QGIS? I have a polygon layer, but ideally need the centrelines as a separate file. Prove that the derivative of $x^w$ is $w x^{w-1}$ for real $w$ Can anyone give a proof of the derivative of this type of function? Specifically showing that  $\dfrac{d(x^w)}{dx} = wx^{w-1}$ for a real $w$?  I tried to use the Taylor series expansion for $(x+dx)^w$ and got the correct result. However, the proof of the Taylor series requires knowledge of the derivative of these functions. So this is essentially circular reasoning. I know that the same series is also given by the binomial expansion, but that's not entirely satisfactory either, because where's the proof that the binomial expansion works for all reals (isn't it only apparent for integers)? So far all of the arguments I've come across involve circular reasoning. I was thinking of showing that the binomial expansion is true for all reals using some form of proof by induction e.g. something like this.  I'm really not sure. Determine which of $N$ points is not on $\sin(ax + b)$, where $a$ and $b$ are unknown. Suppose $N$ points ($(x_1,y_1), (x_2,y_2), ... (x_N,y_N)$) are given from a curve $y=\sin(ax+b)$ where $a, b$ values are unknown. Before giving these $N$ points to you, $y$ coordinate of one point is randomly tampered so that it does not lie on the curve.  Write a program to determine which point among $N$ points is NOT on the sinusoidal curve, whose $a$, $b$ values are unknown.  Any logic on how to approach this question, would be highly appreciate would be very thankful! Run a command in a shell and keep running the command when you close the session I am using Putty to connect to a remote server. What I want to know is if there is any way to write my commands and allow them to keep running after I close the session with Putty. The reason for this is that I do not want to keep the computer ON all the time. Is there any way to do this?.  Update with the solution  For my question as it is presented the best solution is use one of the commands provided such as nohup, because you do not have to install any additional software. But if you are in the same problem use screen, install it and use it. It is amazing.  I have selected the answer of Norman Ramsey as favourite because propose several solutions using commands and screen. But please check the other answers specially the one of PEZ, then you get an insight of what screen is able todo. If n is such that every element $(\mathbb{Z}/n\mathbb{Z})^{\times}$ is a root of $x^2-1$. Prove that $n$ divides 24. How to render a pixelated, occluded wireframe with freestyle? I'd like to combine these features:    The pixelated look of a wire material with anti-aliasing disabled. Proper occlusion of hidden lines.   A wireframe material produces the first feature, freestyle produces the second. Can a line set be created that produces neither double pixels nor gaps? Line thickness alone does not seem to be the answer.  There is the trick with two materials and z-offset, but I have trouble getting a really clean solution at the outline of the object. So a freestyle solution would be best.    The main problem is that freestyle (to the right) produces double pixels: How is it the Voyagers are a few seconds closer to Earth than earlier? The  of March 01, 2013 put it's distance at 14 hrs 04 mins 23 secs of light-travel time from Earth. A  (earlier today) tweet says it is 14 hrs 04 mins 22 secs of light-travel time from Earth. An  tweet put it still further out at 14 hrs 04 mins 25 secs of light-travel time.  The Voyager 2 tweets on Voyager 1's proximity to Earth too has reduced similarly to  earlier today from  reported on Feb 28 2013  It could quite simply be a typo ... If not a typo then my curiosity is piqued; I was under the impression the Voyagers are outbound from the Solar System now without any major in-system activities now.   How is it the Voyagers are closer to Earth than before? Find Items in one column that are not in another column I have two columns in Excel, and I want to find (preferably highlight) the items that are in column B but not in column A.   What's the quickest way to do this?