I am aware of he uniform distribution density function however I am not entirely sure how I would find out the distribution of X.  Thanks very much, any help or hints in the right direction would be gladly appreciated.
For $X$ and $Y$ random variables; $X$ follows the uniform distribution.  (1): if $Y=-\log X$  (2): then it can be shown that $-\log X$ is distributed as $\exp(1)$ {i.e. exponential with mean 1}.  Why is this so? Intuitively statement (2) make sense to me. But i'd like a mathematical proof.  -Probably wrong working:  (1) seems to imply $\exp(-Y) = X$ (which is like saying $X$ is exponentially distributed, which is a contradiction, since its actually uniform!); or is it incorrect for me to do this since $X$ and $Y$ are random variables?  Ultimately how do I prove (2)?  Thanks
This is a  about avoiding outgoing mail being classified as spam.   Also related:                    I'm wondering how to prevent my emails from my site being marked as spam? I'm using sendmail.  I'm trying to send emails through my ruby-on-rails application. The mails are all written in swedish (if that does make a difference?). I don't know why they keep getting marked as spam.  Are there any things that I can do to minimize the risk? How can I switch between starting a coroutine vs running the routine all at once? I'm generating a level, and I have an optional delay to show the steps level generation one by one.  If my generationDelay boolean is true, I want to use StartCoroutine to spread the generation out over time. If it's false, I want to call the same method but without StartCoroutine.  The problem is that the method is type IEnumerator.  I can make two methods one type of IEnumerator and one not. But I wonder if there is a way to use one method.  In the first script top i did:  public bool generationDelay = false;   Then in a method:  private void BeginGame()     {         mazeInstance = Instantiate(mazePrefab) as Maze;          if (generationDelay == true)         {             StartCoroutine(mazeInstance.Generate(generationDelay));         }         else         {             mazeInstance.Generate(generationDelay);         }     }   Then the other script where the method Generate is: In the top:  public float generationStepDelay = 0.01f;   Then the method Generate:  public IEnumerator Generate (bool generationDelay) {          if (generationDelay == true)             WaitForSeconds delay = new WaitForSeconds(generationStepDelay);         cells = new MazeCell[size.x, size.z];         List&lt;MazeCell&gt; activeCells = new List&lt;MazeCell&gt;();         DoFirstGenerationStep(activeCells);         while (activeCells.Count &gt; 0) {             yield return delay;             DoNextGenerationStep(activeCells);         }     }   I'm getting an error on the line:   WaitForSeconds delay = new WaitForSeconds(generationStepDelay);      "Embedded statement cannot be a declaration or labeled statement"   And an error on the line:  yield return delay;      "The name 'delay' does not exist in the current context"   The reason I want to use a bool variable is that when i'm using StartCoroutine it's kind of slow, even if I change the value of generationStepDelay to 0 it's kind of slow. And if I'm not using StartCoroutine and not using the Generate method as IEnumerator it will work fast.  So i wonder what should I do and how? How to calculate Area Under the Curve (AUC), or the c-statistic, by hand Do you gain current HP when you level up? Before marking this post duplicate or voting to get it closed (the reason for which I simply don't get just because it is a grammar forum after all! If I am seeking recommendation for a book that means I will be coming back someday to the same site for my doubts also provided I get a not so harsh reply!), please read this and help this MBA aspirant.   A little about myself and my research so far. I am a very critical reader of English Grammar and the kind of person who aims at perfection.  Currently, I am keen on taking on English grammar again after a certain gap. Well, I am aiming for MBA, so preparing for entrance exams for the same.  I bought the religiously followed Wren and Martin and started studying it chapter by chapter. I was surprised when I read the pronouns chapter. I studied all the rules given and then started practising but it came as surprise when I found when many of the attempted questions were wrong. The reason simply being the rules for those specific sentences were not given.   For example:     We scored as many goals as they/them?    According to me, the answer should be they, but it's them according to the solution manual. Worse still, the solution manual offers no explanation for this.  So after a chat with my teachers I was suggested to follow the blogs of grammarians and look for any rules for the sentences that have contradictory answers!  I did so and was clarified by  post of pronouns, Rule 5.  Then I saw a book by the same author. I was momentarily happy. It was Momentarily because the reviews on Amazon were too bad about this book. After reading a couple of forums about the book I found a common statement saying that no grammar book is exhaustive.  So, it's okay for me to have more than one. And my simple question is what will be a good and almost exhaustive (if not fully exhaustive) book of English grammar to help get through the subtle usages of English in my management entrances  which is crisp and to the point and yet covers all important rules and syntax, and most importantly having practice tests at the end with solutions and explanations? How to run grep with multiple AND patterns? An acquaintance recently posted an image to FB from his recent trip to the JFK Historic Site asking for an explanation of a visible artifact he states was not visible to the naked eye but was visible through the EVF of his camera and in the final image.    The artifact is a series of bright, nearly-vertical bars and a longer, almost horizontal bar superimposed over the lower part of the door frame and the radiator.  I found a very similar artifact in another image of the same room from a blogger which isn't as compressed:    In both cases the artifact seems to be a vertically inverted image of the post and rail outside the saturated window.  From looking at other images of the room available online I know there is spherical glass ornament topping the table lamp whose shape is visible at the left of the photo.  One response to the OP's question is as follows:     RK: It is lens flare. So, even on an iPhone, the camera lens is made   up of multiple lenses. Each designed to focus the image and make sure   it isn't distorted. Remember a lens bends light.  Sometimes, based on   angle and intensity of the incoming light, some stray light bounces   back and forth while traveling through the various elements that make   up what we think of as a camera lens (some of mine have 16 elements   inside a single lens.) Even the best glass is capable of reflecting   some light instead of passing it through the lens elements to the   sensor.  This is what you are seeing. The railing is overexposed, and   some stray light is bouncing around a bit inside the lens.   My response:     DJ: I don't think it's lens flare or ghosting. Those usually form an   image of starbursts, rings, or circles in a row across the image or in   the shape of the aperture; or a veiling glare or haze. This is forming   an image of the light outside minus objects in its path. It is more   likely a direct reflection/projection off a mirror or the glass in a   picture frame elsewhere in the room onto the opposite wall and the   radiator. There are two other clues which seem to confirm this. First,   the image has a gradiated appearance, it is less distinct in the   relatively bright area on the door frame and brighter in the   relatively darker areas behind and on the radiator. You would expect   lens flare to be more evenly specular. Second, the image representing   the light from below the horizontal rail changes angle on the   different faces of the door frame. Unlike a projection, lens flare is   an image created within the lens, the physical shape of the objects it   superimposes do not change its shape.      OP: If that were the case, would I have been able to see said   reflection in the room without the camera? There was no image I could   see with the naked eye. I notice the image in my phone before I took   the pic and did a double, triple and quadruple take before I took it   so I'm sure.      DJ: It would have been visible at the moment the image was taken, but   it may not have appeared nearly as bright and distinct because your   eyes were accommodating for the bright light coming in the window. The   projection could have easily been obscured by you or anyone around you   changing position or moving through the path of the projected   reflection. Also, it's been caught by other people... (see Image 2)   Notice that, in the linked example, the projection is again distorted   by the surface it falls on.   Anyone else want to chime in with an explanation? Which is correct: "__ is different from __" or "__ is different than __"? As someone who learned English later on in life, I was taught that different from is the correct grammar to use: this is different from that. However, it seems these days everyone uses different than instead. I know it's incorrect usage, but does the language evolve if the majority wins?  Edit: Some commented different than is American English and different from is British English, both are OK. IMO, this isn't like color versus colour, though.   Than is used after a comparative adjective, e.g. taller than, whiter than. The word different isn't a comparative adjective, unless used in more different than. Logically, it makes no sense to say different than. Can't save settings after copying site to new server Will everything eventually be a single black hole? Given the idea that gravity is always a factor (even though greater distances are excessively tiny), would everything eventually end up forming a single black hole?  My thinking is that, with infinite time, all mass will eventually be pulled together. I'm wondering whether I should use the present tense for a descriptive statement but in a past context. For example:     (1) I went to the zoo and saw a giraffe last week.       (2) A giraffe is a mammal with long neck.       (3) It is the first time I see that animal with my own eyes.   Statement (2) is a description of a giraffe in general, not the specific one in the zoo, so is it correct to use is?  Statement (3) is an event that is still true now, so should I use is and see?  I'm not native speaker so forgive me if the question is very obvious to others. Prove $\gcd(a,b) \gcd(a,c) \gcd(b,c) \,\text{lcm} (a,b,c)^2=$ $\text{lcm}(a,b)\,\text{lcm}(a,c) \,\text{lcm}(b,c) \gcd(a,b,c)^2$ I'm trying prove $$\gcd(a,b) \gcd(a,c) \gcd(b,c) \,\text{lcm} (a,b,c)^2=\text{lcm}(a,b)\,\text{lcm}(a,c) \,\text{lcm}(b,c) \gcd(a,b,c)^2$$  It is a part of solution of another problem I try a lot to prove it but I could not be succeed.  I tried using exponent manipulation in prime factorizations but it didn't help me. If $\sum_na_n=\infty$ and $a_n\downarrow 0$ then $\sum\limits_n\min(a_{n},\frac{1}{n})=\infty$ If $\sum_na_n=\infty$ and $a_n\downarrow 0$ then $\sum\limits_n\min(a_{n},\frac{1}{n})=\infty$     My  idea:  $$b_{n}=\min(a_{n},\tfrac{1}{n})\Longrightarrow b_{n}\le a_{n},b_{n}\le\tfrac{1}{n}$$ then I don't know how to go on. Given $|z|&lt;1$ with $z \in \mathbb{C}$ prove $\lim_{n \to \infty} z^n=0$.  My attempt given $\epsilon &gt;0$ we wish to show that there exists $\delta &gt;0$ such that $n&gt; \delta$ means $|z^n-0| &lt; \epsilon$.  For $n \in \mathbb{N}$. We have $|z^n|=|z|^n$. We wish $|z|^n &lt; \epsilon$. This means,  $$n \ln |z| &lt; \ln \epsilon$$  But because $\ln |z|&lt;\ln 1=0$ dividing both sides by $\ln |z|$ gives:  $n&gt; \frac{\ln \epsilon}{\ln |z|}$  So that's what we choose for $\delta$.   My question is is how can we get away from the assumption that $n \in \mathbb{N}$. Or is this correct even with the assumption? I have just updated to Minecraft 1.7.4 and now all of a sudden there are barely any mobs spawning.  I checked to make sure that I was in normal difficult and I saw 1 creeper but that is all.  Also I tried different difficulties. Another thing - mobs seem to to spawn normally in the nether and all other mobs except aggressive mobs spawn like they should.  Please if you know how to fix this, please help! Adapt BibTeX style to give all initials I'm using a modification of the plainyr BibTeX style. Although my bib-file contains all initials for some authors (e.g. B.D. McKay), I end up with just the first initial in my document (e.g. B. McKay). I thought about editing the BibTeX style myself, but after seeing the code, I decided against it. Maybe there are some people who are more knowledgeable in this area of expertise.  I want my final document to include the initials for all the names, and not just those of the first name.  This is the relevant code of the BibTeX style, or at least, I think it is the relevant code.  FUNCTION {format.names} { 's :=   #1 'nameptr :=   s num.names$ 'numnames :=   numnames 'namesleft :=     { namesleft #0 &gt; }     { s nameptr "{f.~}{vv~}{ll}{, jj}" format.name$ 't :=       nameptr #1 &gt;         { namesleft #1 &gt;             { ", " * t * }             { numnames #2 &gt;                 { "," * }                 'skip$               if$               t "others" =                 { " et~al." * }                 { " and " * t * }               if$             }           if$         }         't       if$       nameptr #1 + 'nameptr :=       namesleft #1 - 'namesleft :=     }   while$ } What is the difference between linear equation and linear regression? I am using linear equation to find the correlation of values of x, y and z  3x + 4y + 2z = 6 4x + 1y + 4z = 17 8x + 2y + 5z = 9   I would like to know the difference between linear equation and linear regression. Is it the same in approach? When are infinities okay? I'm a little confused about how to handle infinities in physics. On the one hand, I always here things like infinities are not physically tenable answers. GR can't answer anything about the origin of the universe because of singularities. Throw away the radial solution that gives an infinite potential when solving the Laplace equation in spherical coordinates. But then there are other times that the infinities are okay. For instance, it is usually okay to set the reference point for your potential at infinity. There are certain times in David Tong's quantum field theory lecture notes where he seems to almost to hand wave away infinities. So, philosophically, physically, and mathematically when are infinities okay, and when are they not okay? I tried to use the usb-creator.exe from Ubuntu 10.10 desktop i386 ISO image and it does create an USB stick that doesn't boot. When I try to boot from it I get:     Operating System not found.   I try to do this from Windows 7 x64. I'm not looking to create a live Ubuntu media, only an installation USB disk.  I repeated the process several times and I even tried to reformat the USB drive (FAT32), no success yet.  The USB stick is a 8GB one. Gravitational field intensity inside a hollow sphere