Exercise 2.2.5 Analysis 1 Terence Tao Mat. Induc.
Proof of Strong principle of Induction (T. Tao Analysis I)
I understand the proof that  $$Var(aX+bY) = a^2Var(X) +b^2Var(Y) + 2abCov(X,Y), $$  but I don't understand how to prove the generalization to arbitrary linear combinations.  Let $a_i$ be scalars for $i\in {1,\dots ,n}$ so we have a vector $\underline a$, and $\underline X = X_i,\dots ,X_n$ be a vector of correlated random variables. Then $$ Var(a_1X_1 + \dots a_nX_n) = \sum_{i=1}^n a_i^2 \sigma_i^2 + 2 \sum_{i=1}^n \sum_{j&gt;i}^n a_i a_j \text{ Cov}(X_i,X_j)$$  How do we prove this? I imagine there are proofs in the summation notation and in vector notation? How does SO's new auto-login feature work? SO and SU just told me I've been logged in. I'm guessing because I'm already logged into my Google account... but how does SO know that? How to remove chromium-browser settings from Ubuntu 14.04? I have already gone through   and I'm really not getting why the solutions given there are not working for me.  What I'm doing is:   sudo apt-get purge chromium-browser   rm -rf ~/.config/chromium     Then reinstalling by running the following command:   sudo apt-get install chromium-browser    This must remove the settings of chromium-browser and do a fresh installation. The browser is uninstalling, but after being installed it is containing the previous settings, moreover it's not signing me out of even Gmail! How to create a sequence with leading zeroes using brace expansion When I use the following, I get a result as expected:  $ echo {8..10} 8 9 10   How can I use this brace expansion in an easy way, to get the following output?   $ echo {8..10} 08 09 10   I now that this may be obtained using seq (didn't try), but that is not what I am looking for.  Useful info may be that I am restricted to this bash version. (If you have a zsh solution, but no bash solution, please share as well)  $ bash -version GNU bash, version 3.2.51(1)-release (x86_64-suse-linux-gnu) Underscores in words (text) How can I produce the text Word_one_two in LaTeX?  I tried:  Samp\_Dist\_Corr   But, it doesn't quite look right. Also, I want it in the typewriter font, so actually, I'm doing:  \texttt{Samp\_Dist\_Corr}   I find it looks a bit like the underscore is merging in to the bottom of the "D", but maybe it's just because of the typewriter D? Stitch first and edit later or vice versa? I have a set of three images imported in Lightroom 3.6 which I want to stitch into one panorama using Photoshop's CS5.5 Merge to Panorama tool. As I see it I have two options in doing so.   Doing all modifications (colour, brightness, contrast, etc...) first, syncing the changes between the three photos and then stitch them together. Stitching the photos first and then use Lightroom to apply modification to the whole image.   Which of these two methods is considered to be best practice? Finding the Public IP address in a shell script I can find my IP address using ifconfig or hostname -i command.  But how do I find my Public IP?  (I have a static public IP but I want to find it out using unix command) MVC Model require true Convert object string to JSON How do I start applications automatically on login? What happens when you target a "magical effect" with Dispel Magic? Legacy Debian versions and Bash Shellshock We are running Debian Etch, Lenny and Squeeze because upgrades have never been done in this shop; we have over 150 systems running various Debian versions.   In light of the "shell shock" of this week, I assume I need to upgrade bash.  I do not know Debian so I am concerned.   Can I merely execute apt-get install bash on all of my Debian systems and get the correct Bash package while my repository is pointed at a Squeeze entry. If not, what other course of action do I have? Construct a command by putting a string into a tty I managed to do this  echo -n "command" &gt; /dev/tty1  The letters appear, and the cursor moves, but they are "ghosts" – if you hit Enter, nothing happens (they are not in stdin).  Edit:  In the middle of the below screenshot, you see why I see the use of this. (The line with a red caption, right below the line with a yellow caption.) As it is now, you are not really "editing" the note text; you are just asked to write a new text, which will replace the text of the note you are (not really) editing. Thus, I thought it could be remedied by simply pasting the old text into the tty: if the user hits enter, no modification is made. (This program is in Perl/MySQL, but I thought it would be more interesting to ask for a general solution than "how do I do this in Perl".)    Edit 2:  Here is the Perl code, that uses the C code below (works exactly as intended), as well as a new screenshot – hopefully this will clarify things beyond doubt :) Again, look at the middle of the screenshot, where the edit is made to the note text - this time around, the old text is there, for example if you just wanted to fix a typo, you won't have to retype the entire note text.  my $edit_note_text = $edit_note_data[2]; print BOLD, RED, " new text: ", RESET; system("writevt /dev/tty \"$edit_note_text\""); my $new_text = &lt;$in&gt;; $new_text = fix_input($new_text); my $set_text = "UPDATE notes SET note = \"$new_text\" WHERE id = $edit_note_id"; $db-&gt;do($set_text); My computer boots to a black screen, what options do I have to fix it? Why is Facebook missing from the list of ways to share a page in Safari?  That option used to exist before the latest update.  When I'm directed to the extensions box Facebook is not there. How does a phonon cause two electrons to attract each other and form a cooper pair? We know that like charges repel each other. But my professor claimed that two electrons can attract each other as well. What he said was that due to screening an electron travelling at some speed won't repel another electron, but that they will, in some cases, attract each other due to weak phonon exchange. What does that mean? What is phonon exchange? Do two electrons really attract each other? As of the November 2018 core book updates, what changes were made in the latest printings of the PHB that aren't noted in the errata document? Total number of solutions of an equation What is the total number of solutions of an equation of the form $x_1 + x_2 + \cdots + x_r = m$ such that $1 \le x_1 &lt; x_2 &lt; \cdots &lt; x_r &lt; N$ where $N$ is some natural number and $x_1, x_2, \cdots, x_r, m$ are integers?  Also, what would be the solution in the relaxed case where $x_1, x_2, x_3, \dots, x_r$ could be equal? Python, Unicode, and the Windows console When I try to print a Unicode string in a Windows console, I get an error . UnicodeEncodeError: 'charmap' codec can't encode character .... I assume this is because the Windows console does not accept Unicode-only characters. What's the best way around this? Is there any way I can make Python automatically print a ? instead of failing in this situation? Edit:  I'm using Python 2.5.  Note: @LasseV.Karlsen answer with the checkmark is sort of outdated (from 2008). Please use the solutions/answers/suggestions below with care!!  is more relevant as of today (6 Jan 2016). Is it possible to Pivot data using LINQ? I am wondering if it is possible to use LINQ to pivot data from the following layout:  CustID | OrderDate | Qty 1      | 1/1/2008  | 100 2      | 1/2/2008  | 200 1      | 2/2/2008  | 350 2      | 2/28/2008 | 221 1      | 3/12/2008 | 250 2      | 3/15/2008 | 2150   into something like this:  CustID  | Jan- 2008 | Feb- 2008 | Mar - 2008 | 1       | 100       | 350       |  250 2       | 200       | 221       | 2150