Let $z_1, z_2, z_3,$ ...be a sequence of independent random variables s.t. $P(z_i=1)=P(z_i=-1)=\frac 1 2$. Does this sequence converge almost surely?  I am still bothered by the definition of almost sure convergence. If we define a RV Z($\omega_1$=negative sign)=-1, Z($\omega_2$=positive sign)=1  with sample space {$\omega_1$, $\omega_2$} doesnt this sequence then converge to Z? Since P(lim n→∞ Zi(w))=Z(w)=1? But the answer should be no, so I am confused bout this defintion (apology in advnace as I am new to this.)  I think i know how to use second bernoulli to prove it since sum of probabilities which is 1/2 will be infinite meaning that the sequence does not converge. But would some one please explain the above definition?
Let $z_1, z_2, z_3,$ ...be a sequence of independent random variables s.t. $P(z_i=1)=P(z_i=-1)=\frac 1 2$. Does this sequence converge almost surely?  My intuition is telling me that the sequence will approach the random variable Z? i.e. $z_n-&gt;Z$ as $n-&gt;\infty$ and Z is just another RV s.t. $P(Z=1)=P(Z=-1)=\frac 1 2$.  Here is my confusion, if a sequence converges almost surely, then $z_i$ should be different for each term s.t. it approaches Z in the end; but in this sequence $z-i$ is the same for every term. So I dont udnersntand.
SQL Server, convert a named instance to default instance? I need to convert a named instance of SQL server 2005, to a default instance.  Is there a way to do this without a reinstall?    The problem is, 2 out of 6 of the developers, installed with a named instance. So its becoming a pain changing connection strings for the other 4 of us. I am looking for the path of least resistance to getting these 2 back on to our teams standard setup.    Each has expressed that this is going to be, too much trouble and that it will take away from their development time. I assumed that it would take some time to resolve, in the best interest of all involved, I tried combing through configuration apps installed and didn't see anything, so I figured someone with more knowledge of the inner workings would be here. How do you solve a recurrence with a summation function inside: $t(n) = 1 + \sum\limits_{j=0}^{n-1} t(j)$ Show that  $$t(n) = 1 + \sum_{ j=0}^{n-1} t(j)$$  is the same as  $$t(n) = 2^n$$  Initial condition $t(0) = 1$ When my phone MOTOROLA G3(2015) is updated to Marshmallow, its not showing files when connected to PC. While connecting, it shows options as charing only, PTP, MTP, MIDI. When I select MTP, it shows the external card, but not showing the contents, where as internal memory is detected and all files are shown. What are tag badges? How do they work? Difference between static class and singleton pattern? Functional Analysis. Continuous extensions of linear functionals. I would like to show that a continuous functional on a subspace $W$ of a normed space $(V,\|\cdot \|)$ has a unique continuous extension to $V$ iff $W$ is dense in $V$.  I have proved $(\Leftarrow$). But the converse is currently eluding me! Help would be much appreciated! What are best practices for blocking UI when application is busy literally as a hyperbole Possible Duplicate:          I have seen a lot of backlash in internet media against people using the word literally to mean something not literal. Something like "he was literally as big as a house" to mean someone was very big I think this comic sums up the anti literal movement well   Is this actually grammatically incorrect or can the word "literally" be a hyperbole strengthening a metaphor? unable to printout arrayOfInt initialised within the class constructor How to set static IP address? Drush 6 works great of corse on a Drupal 6 website, but will I have all of the Drush 6 features when working on a Drupal 7 website? Should I use Drush 7 when working on a drupal 7 website?  I found this post that says you can switch between the two version easily, is this correct?  switch to drush6 command: composer global require drush/drush:6.*  switch to drush7 command: composer global require drush/drush:dev-master System freezes completely with Intel Bay Trail My system freezes completely at random, frequent intervals. I started to have the same problem in Ubuntu 14.04 but after recent upgrade to 16.04 there is no improvement, in fact it seems worse.  When it happens, it's impossible to do anything. I've tried everything in this thread:  but nothing works, I have to hard reset. I have read all the system logs and journalctl but there is never any information that could help diagnose the problem.  This is a dual-boot system with Windows 10 and there's no problem there, so it's not defective hardware.  My laptop has an Intel Bay Trail processor (Pentium N3540) Is there a way to input particular sections of source code using \lstinputlisting (from the listings package) in a flexible way?  It's possible to use the firstline and lastline options to input particular line ranges, but this isn't robust to changes to the source file.  For example, suppose that the input source file is:  int square(int x) {   return x * x; }  int cube(int x) {   return x * x * x; }   In my LaTeX document, I can include the cube procedure using the command \lstinputlisting[firstline=5,lastline=7]{source.c}.  But suppose I later decide to rewrite the cube procedure.  The input source file becomes:  int square(int x) {   return x * x; }  int cube(int x) {   int x2;   x2 = square(x);   return x * x2; }   Now my LaTeX document will not show the final two lines of cube because the listing uses only lines 5-7.  I could manually update the line range, but this is tedious and error-prone.  Is there a way to mark sections of source code with some kind of label so that the listing will be robust to these sorts of changes?  For instance, perhaps the input source file would be something like  int square(int x) {   return x * x; }  /**\beginlabel{lst:cube} **/ int cube(int x) {   int x2;   x2 = square(x);   return x * x2; } /**\endlabel{lst:cube} **/   so that I could use something like \lstinputlisting[lines=lst:cube]{source.c} to include the section that I want and remain robust to changes.  I am open to using other packages for source code listings (e.g., minted) if helpful. Which Google Updater is the one that I should allow to run? Without appealing to the axiom of choice, it can be shown that (Proposition:) if $X$ is well-orderable, then $2^X$ is totally-orderable.  Question: can we show the stronger result that if $X$ is well-orderable, then so too is $2^X$?  Proof of Proposition. Pick any well-ordering of $X$. Then the lexicographic order totally orders $2^X$.  More explicitly: for any two $f,g \in 2^X$, define $f &lt; g$ iff   there exists $x \in X$ such that $f(x) \neq g(x)$, and if $x \in X$ is minimal such that $f(x) \neq g(x)$, then $f(x)=0$ and $g(x)=1$.   It can be shown that $&lt;$ totally-orders $2^X$. Changing the homepage Powering a WS2812B strip Let $X$ be any distribution with defined mean, $\mu$, and standard deviation, $\sigma$.  The central limit theorem says that  $$ \sqrt{n}\frac{\bar{X} - \mu}{\sigma}  $$ converges in distribution to a standard normal distribution.  If we replace $\sigma$ by the sample standard deviation $S$, is there a theorem stating that  $$ \sqrt{n}\frac{\bar{X} - \mu}{S}  $$ converges in distribution to a t-distribution?  Since for large $n$ a t-distribution approaches a normal, the theorem, if it exists, may state that the limit is a standard normal distribution.  Hence, it would seem to me that t-distributions are not very useful - that they are useful only when $X$ is roughly a normal.  Is this the case?    If it is possible, would you indicate references that containing a proof of this CLT when $\sigma$ is replace by $S$?  Such a reference could preferably use measure theory concepts.  But anything would be great to me at this point. Standard normal distribution is truncated to the left by 2 Would you please show me how to write the pdf distributions of the following   Standard normal distribution is truncated to the left by 2  Exponential distribution is truncated to the left by 2  Cauchy distribution is truncated to the left by 2  i do not know what does mean by truncated to the left by 2 All of my Firefox add-ons have been disabled suddenly, how can I re-enable them?