I moved to close this. The question has been aksed and answered before:  Suppose you have a Poisson process with intensity $\lambda$, and let $N(t)$ be the number of events that have occurred at time $t$. I can prove, and have seen many references on the internet, for the fact that $\mathbb{E}[N(t)] = \lambda t.$ I'm trying to find a proof for the fact that $$\lim_{t \to \infty} \frac{N(t)}{t} = \lambda$ almost surely. I saw this once, but cannot find a good reference.
Let $N = \{N(t) \}_{t\geq 0 }$ be a Poisson process. I already know that $N(t)- \lambda t$ is a martingale where $\mathbb{E} [ N(t) ] = \lambda t$.   I want to prove that $$ \frac{N(t)}{t} \rightarrow \lambda, \quad t \rightarrow \infty \quad \text{ a.s.}. $$  To prove this I received a hint that I should use the martingale law of large numbers which states that:  Theorem: (Martingale LLN)  Let $S= (S_n = \sum_{k=1}^n X_k)_{n=1,2,\ldots}$ be a martingale with respect to the filtration $(\mathcal{F}_n)_{n=1,2,\ldots}$. Then $\sum_{k=1}^n X_k/k$ converges almost surely on the set $\left\{ \sum_{k=1}^\infty k^{-2} \mathbb{E}\left[ X_k^2 \mid \mathcal{F}_{k-1} \right] &lt; \infty \right\}$. Hence $S_n/n \rightarrow 0$ a.s. on the same set.  Now to use this theorem I think I need to use Doob-Meyer to decompose $N(t)$ as the sum of a martingale and an increasing process. Which would be the martingale $M(t) = N(t) - \lambda t$ and increasing process $A(t) = \lambda t$. But how is $M(t)$ the sum of a martingale? And I do not know how this exactly relates to the thing I have to prove. Thanks for any help.
Is it consistent with ZF for there to exist a set $S$ such that the power set $P(S)$ is countable? If so, what is the weakest form of the axiom of choice needed to prove that no such set exists? How should I act if I discover a user plagiarizing fellow users' posts? What if I find someone plagiarizing external resources without attribution?  See also: I have a WordPress install on a dedicated server with decent memory (upwards of 8GB).  The WordPress install includes many plugins - 50 active plugins. Most important to note is the  plugin to make WordPress multilingual. The theme is .  The stack is LAMP - CentOS 6.3, Apache 2.4.4, Percona MySQL, PHP-FPM.  I have been quite stupid, and didn't set up a test or staging server.  Everything was running fine till I updated to WordPress 3.8, and also updated several plugins at the same time.  I didn't pay attention at the time, but all my widgets got messed up, and so did any sort of navigation etc.  I enabled error_log for a bit, i.e.  ini_set('log_errors',TRUE); ini_set('error_reporting', E_ALL); ini_set('error_log', dirname(__FILE__) . '/error_log.txt');   This resulted in an immense file - basically the error_log grows by about a GB every 4 minutes or so.  The primary error I found in the error log was database server disconnect, better known as      MySQL server has gone away   The first priority was to patch the problem. So I installed  plugin. This mostly, though not completely solved the issue. Many of the widgets came back.  However, the special (non-English) characters are all messed up (converted to question marks). I tried to fix this by opening wp-mysql-ping.php and commenting out   if ( !DB_CHARSET &amp;&amp; version_compare(mysql_get_server_info($this-&gt;dbh), '4.1.0', '&gt;='))                                     {                                         $this-&gt;query("SET NAMES '" . DB_CHARSET . "'");                                     }   Didn't help.  Besides, the plugin is not a permanent or complete solution. So leaving the site in this messed up state, I started working on other possibilities:  As suggested at , I edited my.cnf to add  wait_timeout = 60   Didn't help.  Also added caching to my.cnf  query_cache_type = 1 query_cache_limit = 8M query_cache_size = 512M   Didn't help.  (and yes, I restarted mysql every time I edited my.cnf)  Also tried the method : In wp-db.php before  $this-&gt;select( $this-&gt;dbname, $this-&gt;dbh );   I added  $this-&gt;query("set session wait_timeout=600" );   Didn't help.  I'm at my wit's end. Please help! What are Java command line options to set to allow JVM to be remotely debugged? I know there's some JAVA_OPTS to set to remotely debug a Java program.  What are they and what do they mean ? failed to fetch security undermined error [IP: 91.189.91.38 80] in ubuntu 20.04 Types of things vs. types of thing What are the best methods to show a data table in mobile / tablet view? What is the 64-bit Firefox Beta PPA? I recently discovered that my computer is 64-bit. I have backed up my Home folder, and reinstalled Ubuntu. The reinstall wasn't nearly as painful as I thought. There is one thing that I can't quite seem to figure out: how do I get the 64-bit Firefox Beta build? I always get the Beta builds, but I want to take advantage of the 64-bit architecture of my computer.  says that Mozilla has come out with a 64-bit version of Firefox, but I can't seem to find it.  I do understand  of using a 64-bit browser, but I've decided to jump right in and do it anyway. (Flash and Java are already 64-bit, and who cares about Silverlight, since it's not for Linux anyway?)  There's only one issue, and it's a big one: I can't find the 64-bit Beta PPA!!! (I really hate using .tar.gz files, but I'd be willing to do that as long as I could still access Firefox via the Launcher. Oh, speaking of which, I don't understand .tar.gz files. Once, I managed to run one (the Dropbox Beta build,) but I have no idea whatsoever on how to install them: as in, click on the icon and go.) Executing cmd.exe commands from Java Current situation: I'm required to cite "normative" documents (read: standards and manuals) separately from articles and books etc. Using biblatex and biber with the numeric style I'm doing that by issuing multiple printbibliography commands like so:  \usepackage[style=numeric,backend=biber,bibencoding=utf8]{biblatex} %... lots of stuff \printbibliography[type=standard,title=Standards \&amp; Manuals] \vskip\baselineskip \printbibliography[type=manual,heading=none] \printbibliography[nottype=standard,nottype=manual]   which gives a heading Standards &amp; Manuals, prints a list of manuals, leaves a bit of space then the list of standards and finally, under a new heading Bibliography gives books and articles etc. Thus far everything is fine, enter:  The Problem: I'm using the shorthand field in my database for the standards and manuals which works perfect as far as output within the text is concerned and sets the shorthand as the label in the bibliography and prints the details to the right of it - problem is: my shorthands are considerably longer than the labels generated by the numeric style, but even though they're in three different \printbibliography commands, everything gets indented to the maximum width of the label, making the part for books and articles look quite ridiculous.  The following shows the problem with only two separate \printbibliography (and an obviously exaggeratedly long shorthand):   Now  what I would obviously want is for each of the \printbibliography commands to only use the indentation actually necessary to accommodate the largest label appearing within, i.e. rendering the Manuals section as above, but the Bibliography section like follows:   Can anyone help me to achieve this? Or give me some pointer where to look?  The above was produced with the following MWE on an up-to-date TeXLive 2014:  \documentclass[10pt,british]{scrbook}  \usepackage{babel} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern}  \usepackage{filecontents} \begin{filecontents*}{\jobname.bib} @manual{HowTo,     title = {How to Make the World Go Round},     shorthand = {HowToMakeTheWorlGoRound},     publisher = {The Publisher},     year = {2000} }  @article{Article,     author = {Author, A. and Buthor, B.},     journal = {Journal Title},     pages = {113--126},     title = {The Article},     volume = {65},     year = {1900} } \end{filecontents*}  \usepackage[style=numeric,backend=biber,bibencoding=utf8]{biblatex} \addbibresource{\jobname.bib}  \usepackage[hidelinks]{hyperref}  \begin{document} \let\cleardoublepage\relax \chapter{The Document} Nothing to see here, but we refer to the \parencite{HowTo} based on a principle discovered by \textcite{Article}.  \printbibliography[type=manual,title=Manuals] \printbibliography[nottype=manual] \end{document} Back transforming regression results when modeling log(y) I'm fitting a regression on the $\log(y)$.  Is it valid to back transform point estimates (and confidence/prediction intervals) by exponentiation?  I don't believe so, since $E[f(X)] \ne f(E[X])$ but wanted other's opinions.  My example below shows conflicts with back transforming (.239 vs .219).  set.seed(123)  a=-5 b=2  x=runif(100,0,1) y=exp(a*x+b+rnorm(100,0,.2)) # plot(x,y)  ### NLS Fit f &lt;- function(x,a,b) {exp(a*x+b)}  fit &lt;- nls(y ~ exp(a*x+b),  start = c(a=-10, b=15))  co=coef(fit) # curve(f(x=x, a=co[1], b=co[2]), add = TRUE,col=2,lwd=1.2)  predict(fit,newdata=data.frame(x=.7)) [1] 0.2393773  ### LM Fit # plot(x,log(y)) # abline(lm(log(y)~x),col=2) fit=lm(log(y)~x) temp=predict(fit,newdata=data.frame(x=.7),interval='prediction') exp(temp)         fit       lwr       upr 1 0.2199471 0.1492762 0.3240752 Does a man contain all the genes needed to make a woman? This question is brought on by a Sci Fi novel I am thinking about writing. The plot device involves a colonist in charge of building a population on a new planet who loses his supply of embryos and so forth in the landing. With the artificial "wombs" intact, he populates the planet with clones of himself.  If he wanted to introduce enough variation (without making up new genes, only rearranging) in the population to allow them to be able to reproduce naturally without inbreeding problems, would he be able to? In particular, could he make a woman?  Alternately, could a woman (without a Y chromosome) do the the inverse?  At the risk of introducing bias into the answers, the plot as it stands assumes he is a male and he CAN'T do any of the above and he is restricted to clones. In the sentence-     You must wear a suit to an interview   shouldn't the to be replaced by for? Or what's the difference between these two here? If $(X,d)$ is a complete metric space and $A$ is closed then show that for $x \in X$ there exists an element $a_0 \in A$ such that $d(x,A)=d(x,a_0)$ Converting from DMS to DD using Python in Field Calculator? Given a file like this:  1,768,12,46576457,7898 1,123,435,134,146 2,345,6756856,12312,1311 5,234,567465,12341,1341 1,3245,4356345,2442,13 9,423,2342,121,463 9,989,342,121,1212   I would like to list all rows (in bash terminal) such that the value in column 1 appears at least twice (in column 1). The result should be  1,768,12,46576457,7898 1,123,435,134,146 1,3245,4356345,2442,13 9,423,2342,121,463 9,989,342,121,1212 Generating hatched filling using Region functionality What solutions does the equation $4x\equiv2\mod10$ have? (Hint, it will have more that one.) What about solutions to an equation $ax\equiv d\mod n$, where $d=\gcd(a,n)$?  This is the question that appeared in our universities last year paper. But it seems very difficult for me like for first part I am able to solve by simplifications I got $x=5\cdot Z +3$ as solution which makes sense and seems fair but for the second one, I have no idea how can I approach that. Can you please help me out with that one. I tried to simplify that by using Euclidean algorithm but still, I didn't get any idea that what actually the solution will be. My final exam is on 2nd of next month and because of covid I am unable to contact in person for any help. So any help from this site will be appreciated. Thank you for looking at this question! SELECT * EXCEPT Is there any RDBMS that implements something like SELECT * EXCEPT?  What I'm after is getting all of the fields except a specific TEXT/BLOB field, and I'd like to just select everything else.  Almost daily I complain to my coworkers that someone should implement this...  It's terribly annoying that it doesn't exist.  Edit: I understand everyone's concern for SELECT *.  I know the risks associated with SELECT *.  However, this, at least in my situation, would not be used for any Production level code, or even Development level code; strictly for debugging, when I need to see all of the values easily.  As I've stated in some of the comments, where I work is strictly a commandline shop, doing everything over ssh.  This makes it difficult to use any gui tools (external connections to the database aren't allowed), etc etc.  Thanks for the suggestions though. In the Law of Reflection, the angle of incidence is equal to angle of reflection. Why is this true? This is clearly true experimentally, but how does one prove this true mathematically?