Let $X_1,X_2,...$ denote iid random variables such that $X_1$ is continuously distributed with density $p(x)$ and distribution function $F(x)$ where $F(0) = 0, F(x) &gt; 0$ for all $x &gt; 0$, and $p(0) &gt; 0$. Let   $\hspace{15mm}Y = n\cdot  \min \{X_1,...,X_n\}$.  Then there exists a random variable Y such that:   $\hspace{15mm} Y_n \xrightarrow{D}Y$ as $n \xrightarrow{} \infty$.  What is the distribution function of $Y$?  I'm not sure how to proceed with this question. Any insight would be much appreciated.
Let $Z_i,Z_2,\ldots$ be IID Random Variables with density $f$. Suppose that $P(Z_i&gt;0)=1$ and that $\lambda=\lim_{x \to 0+} f(x)&gt;0$. How can I show that $X_n=n \times \min\{Z_i\}$ has a limiting exponential distribution with mean$1/\lambda$?  I know that the CDF of the first order Statistic is for $t&gt;0$  $$F_{min \{ Z_i \}} (t)= 1- \left[ 1-F(t) \right]^{n}$$  Convergence in distribution requires that $F_n (t) \to F(t)$ for some distribution function $F$ but I cannot quite take the limit of the above, with $t$ replaced by $x/n$, as the distribution is not known. I guess I have to insert the second piece of information somewhere but I do not recognise where.  Could you please give me a hint or two?  Thank you.
I am trying to help a friend who is very concerned that she has just lost all of the data on her 32GB SanDisk microSD card.  I told her I would post this and see if anyone could offer help.  Here are the details:   The SanDisk microSD card is less than two years old.  Probably only about a year old.  The store will take it back, but that doesn't really help.  She wants her data. She used the card in her Samsung Galaxy S4 smartphone for over a month.  The phone was never exposed to water or temperature extremes. Today, her phone suddenly said something like "Your SD card is blank or an unsupported format". Her phone was then unable to read the SanDisk microSD card at all.   Here's what she has tried:   Rebooting her phone.  Had no effect. Taking out the SanDisk microSD card and putting it inside a full-size SD card adapter, and inserting it into a Windows 7 SP1 computer.  The computer did not recognize its presence at all. Putting the SanDisk microSD card into a USB 2.0 external card reader and inserting the card reader into a USB port on a Windows 7 SP1 computer.  The computer recognized the card reader and installed drivers for it.  The card reader shows up as 'Removable Disk' in Windows Explorer.  Clicking on 'Removable Disk' in Windows Explorer results in the error "Insert disk: Please insert a disk into the Removable Disk". After trying #3, tried opening the 'Removable Disk' in FreeCommander.  Doing so results in the error "The device is not ready".   She does not have access to a linux box, but does have access to her Android phone.  What can she do to recover the data on her 32GB SanDisk microSD card? About the absence of metal in machinery and buildings of the past Optimal estimation of a single parameter given data at different scales Let $A,B,C,D$ be independent normal random variables distributed according to $\mathcal{N}(0,\sigma^2)$ and $E = A+B+C+D$.  The goal is to provide an optimal estimate of $\sigma^2$ given samples of $A,B,E$ but not $C,D$.  In this case optimal refers to minimising the root-mean square deviation of the estimate $\hat{\sigma}^2$:  $$RMSD(\hat{\sigma}^2) = \sqrt{\mathbb{E}\left[\left(\hat{\sigma}^2-\sigma^2\right)^2\right]}$$  I can think of a simple approach to estimate $\sigma^2$ given samples of $A,B$ which is to simply compute the sample variance. Also I know that the variance of $E$ is $4\sigma^2$ so I could compute the sample variance of $E$ and divide it by 4 to get an estimate of $\sigma^2$. However it should be possible to combine the information from both $E$ and $A,B$ to improve estimation $\sigma^2$. Could you outline the approach to use both sets of information? "Abusing" bounties for unlimited profit Six days ago I put a +50 bounty on  which wasn't asked by me, but to which I'd previously posted a pretty decent answer. Since then I've earned 130 reputation from upvotes on my answer, netting me +80 reputation and a bronze badge from no work.  It seems possible to slowly generate free reputation by "investing" in your answers like this. I expect I could repeat this perpetually and continue to profit from it.  This strikes me as an abuse of the bounty system. Should anything be done to prevent it?  EDIT: For more concrete questions:   If I put another bounty on this question after this one expired, would that be considered abuse? What if I did it several times? ( the answer was  "yes", but the author was doing so to promote his own project.) What if I didn't do it to the same question, but was constantly promoting questions I'd answered, in order to promote my answers? What tense should I use to describe something in the past but probably is still true? The title may be a bit confusing. See the below example:     The movie (was/is) so good.   The simplest guideline on deciding which tense to use, past tense or present tense, to describe an object comes down to whether the description was only true that that time or is true even to the present point of time.  But now we talking about literary work(movies,books,etc) of which quality tend to stay the same over time, which tense should we use? Could humans choose to establish fibre digesting colonies in our guts? Projecting sp objects in R I have a number of shapefiles in different CRSs (mostly WGS84 lat/lon) that I'd like to transform into a common projection (likely Albers Equal Area Conic, but I may ask for help on choosing in another question once my problem gets better-defined).  I spent a few months doing spatial stats stuff in R, but it was 5 years ago.  For the life of me, I cannot remember how to transform an sp object (e.g. SpatialPolygonsDataFrame) from one projection to another.  Example code:  P4S.latlon &lt;- CRS("+proj=longlat +datum=WGS84") hrr.shp &lt;- readShapePoly("HRR_Bdry"), verbose=TRUE, proj4string=P4S.latlon)  # Shapefile available at  #   http://www.dartmouthatlas.org/downloads/geography/hrr_bdry.zip  #   but you must rename all the filenames to have the same  #   capitalization for it to work in R   Now I have a SpatialPolygonsDataFrame with appropriate projection information, but I'd like to transform it to the desired projection.  I recall there being a somewhat unintuitively-named function for this, but I can't remember what it is.  Note that I do not want just to change the CRS but to change the coordinates to match ("reproject", "transform", etc.).  Edit  Excluding AK/HI which are annoyingly placed in Mexico for this shapefile:  library(taRifx.geo) hrr.shp &lt;-    subset(hrr.shp, !(grepl( "AK-" , hrr.shp@data$HRRCITY ) |                                      grepl( "HI-" , hrr.shp@data$HRRCITY )) ) proj4string(hrr.shp) &lt;- P4S.latlon When looking at most sites (including SO), most of them use:  &lt;input type="button" /&gt;   instead of:  &lt;button&gt;&lt;/button&gt;    What are the main differences between the two, if any? Are there valid reasons to use one instead of the other? Are there valid reasons to use combine them? Does using &lt;button&gt; come with compatibility issues, seeing it is not very widely used? biblatex: Full Citations in Footnotes on Every Page How can I create this sort of bibliography using biblatex, ie, bibliography at the footer of every page? Varnish cache with Nginx SSL termination How can I use varnish cache with nginx SSL termination for my SSL-Only (Https only) website? I use ubuntu 18.04 and I am currently running a wordpress site on my server. Why do fantasy writers depict pointy hats as the headgear of choice for Witches and Wizards? They don't always wear a hat, but when they do, it's pointy.   Why do fantasy writers depict pointy hats as the headgear of choice for Witches and Wizards? First off I did try the power options setting and it's on 'Never' but that did nothing. The drive still spins down after 15-20 seconds of not being used. I need it to spin all the time because it's more efficient with my way of using it. Also listening to music from that drive is absolutely destroying the drive making it spin up and down all the time causing immense wear and tear. Any ideas on how to stop it from spinning down? Please don't suggest bat files that keep creating editing a file.  &nbsp;  Oh and here is the model of the drive: wdc wd10jpvx-22jc3t0 and my OS is Windows 10 64 bit. Show that $\mathbb{Q}\times \mathbb{Q}$ is denumerable I am new to functions and relations, and with some concepts I am not so familiar. I have a question in an homework:     Show that $\mathbb{Q} \times\mathbb{Q}$ is denumerable.   From what I understood, denumerable means that it is infinitively countable. There are some  on the web about this topic, but I am still not understanding their explanation, maybe because they are not explained so easily (for me).  From what I understood, a set is denumerable if there's a relation with the natural numbers (but I am still not understanding what is this relation).  I have heard also about equinumerous sets (contain a bijective function = onto + one-to-one function), but I cannot relate this information with the problem to try to solve it. Boliviana de Aviacion (BOA) tariff codes I am trying to book one-way tickets on BOA on Jan 13 2015 from La Paz (LBP) to Buenos Aires (EZE) from their website.  Their website asks me to make a choice between 7(!) different tariff codes - they are SSEEOB, QEEOB, TMEEOB, VEEOB, HEEOB, BEEOB, YOW. The prices are different for every tariff, but other than that I cannot see any obvious difference and there is no explanation available. Also I am unable to find anything useful on their website  Could anyone from Bolivia or familiar with BOA explain what is going on there and what choice am I supposed to make? What apps are people using to record voice meetings on the iPad? I'm looking to do some light recording of voice conversations and field recordings on my iPad using the built in microphone.    seems to be the obvious Apple solution, but is too heavy for my needs. By the time I set up a new project, turn off the metronome, etc..., I find myself wishing I just had  running on my iPad.   I don't mind paying for the app, but wondered what people are using for both light voice recording as well as longer and more numerous recording / light editing where some tagging and filing might be possible. In Bash, when to alias, when to script, and when to write a function? Does using the Console Commands negatively affect me? I'm wondering in case I get stuck in an area that I'm unable to get out of or if I clip into a wall and I suddenly can't move. I've had to disable clipping to get out of certain areas in Skyrim before but I want to make sure that things like my achievements and stuff won't shut out on my playthrough.  If I enable something, spawn an item that didn't spawn for me, or just basically use the console command to fix any bugs, what will happen achievement and game-wise? How to tell if a certain item is in a player inventory? I am trying to create a challenge world where a certain item (e.g. a Netherite Sword) must be collected. How can I check automatically if a player has it (using commands), and then congratulate the winner? This question is specific to Bedrock Edition 1.16. I have tried other solutions so many times and they don't seem to work. See also: Reapplying for a UK Standard Visitor visa after a refusal Follow up: instance control in Java without enum