$x_1,x_2,\dots,x_n$ are i.i.d RVs uniformly distributed on $\{1,2,\dots,N\}$. Let $x_1,x_2,\dots,x_n$ be independent identically distributed random variables uniform on $\{1,2,\dots,N\}$, and let: $Y_n:=\text{the number of different elements in } \{x_1,x_2,\dots,x_n\}$.  Let $T:=\inf\{n:Y_n=N\}$.  What is $E\left[T\right]$?
Expected time to roll all 1 through 6 on a die What is the average number of times it would it take to roll a fair 6-sided die and get all numbers on the die?  The order in which the numbers appear does not matter.  I had this questions explained to me by a professor (not math professor), but it was not clear in the explanation.  We were given the answer $(1-(\frac56)^n)^6 = .5$ or $n = 12.152$  Can someone please explain this to me, possibly with a link to a general topic?
Info Icon vs Question Mark How to tell the difference between linear and non-linear regression models? I was reading the following link on non linear regression . My understanding from reading the first section "Nonlinear Regression vs. Linear Regression" was that the equation below is actually a linear regression, is that correct? If so why?  $$y = b_1x^3 + b_2x^2 + b_3x + c$$  Am I also to understand that in non linear regression multicollinearity isn't an issue? I know that multicollinearity can be an issue in linear regression so surely if the model above is in fact a linear regression there would be multicollinearity? LaTeX on Stack Overflow? in questions, answers, and comments. Can we get something like this going on Stack Overflow?  I think it’d be appropriate as I at least pretty often want to write something like n^2 and would benefit greatly from prettier markup. Temporal Table that doesn't use system time We want to implement temporal tables in SQL Server 2016. We are creating a Datawarehouse and developing Type 2 Slowly changing Dimension Tables.  For the BeginDate, we want it to be dependent on transaction date and not the current getdate time. We are reprocessing a history of transactions. Example below, customer has gym or bank status, and goes from inactive, to active, or pending, dependent on transaction date.  We currently have this.  CREATE TABLE dbo.Department (     CustomerId int primary key,     MembershipStatus int,     TransactionDate datetime  );   We we would like to create a table like this.  CREATE TABLE dbo.DepartmentHistory (     CustomerId int primary key,     MembershipStatus int,     TransactionDate datetime,     BeginDatetime datetime,     EndDatettime datetime  );   Example usage would be:   first Customer Transaction on 3/5/2018 as Pending P        +------------+--------+-----------+---------+     | CustomerId | Status | BeginDate | EndDate |     +------------+--------+-----------+---------+     | 1          | P      | 3/5/2018  | NULL    |     +------------+--------+-----------+---------+    second transaction is 4/21/2018 is Active A        +------------+--------+-----------+-----------+     | CustomerId | Status | BeginDate | EndDate   |     +------------+--------+-----------+-----------+     | 1          | P      | 3/5/2018  | 4/21/2018 |     | 1          | A      | 4/21/2018 | NULL      |     +------------+--------+-----------+-----------+ Present perfect for past action with present effect $\varepsilon$-number countability without choice Let $\alpha\mapsto\varepsilon_\alpha$ be the enumeration of the $\varepsilon$-numbers--that is, those $\alpha$ such that $\omega^\alpha=\alpha$--by the ordinals.  If we know that countable unions of countable sets are countable, (slightly weaker than countable choice), then we can show fairly easily that $\varepsilon_\alpha$ is countable iff $\alpha$ is countable.  I think I saw a result at some point that $\varepsilon_0$ is countable without relying on any choice principle (though I can't recall where I saw that). Is this true? More generally, for which $\alpha$ can we conclude that $\varepsilon_\alpha$ is countable in ZF?  Update: I have been able to prove (in ZF) that if $\varepsilon_\alpha$ is countable, then $|\alpha|&lt;\aleph_1$, so that's one direction. I'm still a bit uncertain about the answers I've got so far. Normal text is invisible when using beamer with notes and XeLaTeX I'm using beamer with xelatex to prepare a talk.  If I switch on notes on right I'm experiencing strange behaviour:  When using csqoutes in the title, all normal text afterwards becomes invisible (or white). The text selectable in the output pdf. Note that the text is also visible in the small preview on the note page.  This does not happen when using pdflatex or lualatex.  MWE:  \documentclass{beamer} \usepackage[autostyle=true,german=quotes]{csquotes}  \usepackage{pgfpages} \setbeameroption{show notes} \setbeameroption{show notes on second screen=right}  \begin{document}    \begin{frame}         \frametitle{\enquote{Enqouted Title}}     \end{frame}      \begin{frame}         \frametitle{This is the second slide}         Lorem ipsum dolor sit amet, consectetuer adipiscing elit.     \end{frame} \end{document}   Update:  csqoutes is not the culprit. Seems to be a general problem when using xelatex and beamer with notes enabled.  New MWE:  \documentclass{beamer}  \usepackage{pgfpages} \setbeameroption{show notes} \setbeameroption{show notes on second screen=right}  \begin{document}     \begin{frame}         \frametitle{First Slide Title}         Here be text     \end{frame}      \begin{frame}         \frametitle{This is the second slide}         Lorem ipsum dolor sit amet, consectetuer adipiscing elit.     \end{frame} \end{document} Aliased edge around light source in cycles Often when a mesh light is visible in the camera, the edges are not anti-aliased.  No matter how many samples I do, the lamp border always looks jagged and pixelated.  Is this a rendering error? How can I get around this issue?    And zoomed in 4x: I found many related questions on this site but I am still not sure about the rules. How can I decide if I need a visa to transit? Schengen members as of May 2021 are as follows:  Austria Belgium Czech Republic Denmark (excluding Greenland and the Faroe Islands - but an open border with the Schengen Area is maintained) Estonia Finland France (excluding overseas departments and collectivities) Germany Greece Hungary Iceland Italy Latvia Liechtenstein Lithuania Luxembourg Malta Netherlands (excluding Aruba, Curaçao, Sint Maarten and the Caribbean Netherlands) Norway (excluding Svalbard) Poland Portugal Slovakia Slovenia Spain (except Ceuta and Melilla) Sweden Switzerland How can I programmatically create a new cron job? I want to be able to programatically add a new cron job, what is the best way to do this?  From , it seems I could dump the current crontab and then append a new one, piping that back into crontab:  (crontab -l ; echo "0 * * * * wget -O - -q http://www.example.com/cron.php") | crontab -   Is there a better way? Automatically convert quotations in the form of "abc" to become ``abc" How can I disable landscape mode in Android? Quite a few of the questions given on this site mention a photon in vacuum having a rest frame such as it having a zero mass in its rest frame. I find this contradictory since photons must travel at the speed of light in all frames according to special relativity.  Does a photon in vacuum have a rest frame? Include the "People Reached" and "Helpful Flags (count)" data in SEDE Lowering elevation of lake using QGIS I am in QGIS. I have a raster layer of mountains and a lake. I have a vector layer (shapefile) of the lake. It sits nicely where it supposed to be over the raster level. I am going to convert this height map to an STL so that I can cnc this out of wood, so I'd like the actual lake to be something like 200 feet lower than it shows on the raster layer so that I can fill it with blue epoxy. I have read a lot of clipping and everything else, and I am lost. An inequality involving 4 real or complex numbers Let $A$ be a squared matrix, and suppose there exists an $n\in \Bbb N$ in a way that $A^n=0$. Show that $I-A$ is invertible and that $(I-A)^{-1}=I+A+\cdots+A^{n-1}$    I don't have a clue where to start from. The Density of Clouds How to filter string for unwanted characters using regex? Formula based on picklist I am trying to create a formula that will put a 1 in a hidden field if the value of a picklist is equal to a specific value.  I keep getting stuck here and I can't figure out what I am missing?  IF(ISPICKVAL( Examiner_Type__c ="Pending Examiner Status","1","",))