Expected Value Inequality - bounded by sums of probabilitys I have to show the following inequality: $$ \sum_{k=0}^{\infty} P[Y&gt;k] \leq E[Y] \leq \sum_{k=1}^{\infty} P[Y&gt;k]$$ under the condition that Y is a non negative continuous random Variable.  My first idea was to rewrite the inequality using integrals, but then i end up having to integrate over a sum, which is nothing im particulary excited about. Any Ideas on how i can do this?
Why is $\mathbb{E}[X] = 1 + \sum^\infty_{k=1}\mathbb{P}(X > k)$ true? I'm working through a problem regarding expected values in Markov chains, and at some point it says:     Recall from probability that if $X$ is a positive integer valued random variable, then $\mathbb{E}[X] = 1 + \sum^\infty_{k=1}\mathbb{P}(X &gt; k)$.   I know that by definition $\mathbb{E}[X] = \sum_a a\mathbb{P}(X = a)$, but I can't see how the above equality follows from this, nor am I sure if this is how to approach the problem.
How to Select Hyperlink Text in Google Chrome? I am running Google Chrome 5. How can I select hyper link text using the mouse with, for example, CTRL or ALT keys? Sci-fi novel involving living forever through teleporter copies Remove Actions/Filters added via Anonymous Functions That's a freaking bad practice I must say. Spent last two hours finding a solution to remove actions and filters added via Anonymous functions.    This is the code used on a Parent Theme, and I need to remove it.  /**  * Add custom columns to admin comments grid  *  * Rate that user set.  */ add_filter( 'manage_edit-comments_columns', function( $default ) {     $columns['smr_comment_rate']  = __( 'Rate', 'txtdmn' );      return array_slice( $default, 0, 3, true ) + $columns + array_slice( $default, 2, NULL, true ); });   Got , played with it heavily, but no help. So, is there any other alternative that will remove actions/filters added via anonymous functions?  Thanks I'm writing a script which shows the git log for a directory when I cd into it. Such a log can be overwhelming, containing hundreds of lines. So far I have been limiting that to a hard-coded 20 lines (... | head -n 20), which is fine on the screen at work, but too much on the smaller MacBook screen at home.  I would prefer the log to take up about half the (vertical) screen on either terminal. And "terminal" also changes: it's Gnome terminal at work, but iTerm2 at home. And I do not use screen or tmux.  How do I find the number of vertical lines available in a terminal from command line? given value of matrix, getting it's coordinate Finding minimum-area-rectangle for given points? How to know number of cores of a system in Linux? I wanted to find out how many cores my system has, so I searched the same question in Google. I got some commands such as the lscpu command. When I tried this command, it gave me the following result:  $ lscpu Architecture:          x86_64 CPU op-mode(s):        32-bit, 64-bit Byte Order:            Little Endian CPU(s):                4 On-line CPU(s) list:   0-3 Thread(s) per core:    1 Core(s) per socket:    4 Socket(s):             1 NUMA node(s):          1 Vendor ID:             GenuineIntel CPU family:            6 Model:                 23 Stepping:              10 CPU MHz:               1998.000 BogoMIPS:              5302.48 Virtualization:        VT-x L1d cache:             32K L1i cache:             32K L2 cache:              2048K NUMA node0 CPU(s):     0-3   In particular, this output shows:   CPU(s): 4 Core(s) per socket: 4 CPU family: 6   Which of those indicates cores of a Linux system?  Is there any other command to tell the number of cores, or am I assuming it is completely wrong? Make browser window blink in task Bar How do I make a user's browser blink/flash/highlight in the task bar using JavaScript? For example, if I make an AJAX request every 10 seconds to see if the user has any new messages on the server, I want the user to know it right away, even if he is using another application at the time.  Edit: These users do want to be distracted when a new message arrives. Retagging questions in terms of possible answer technologies When entering the title of this query I was pointed to , which has tentatively endorsed this concept, especially once an answer has been accepted.  Can I get a confirmation that it is actually good practice before an answer has been accepted? (IMHO it is not.)  Two examples:   : I didn't need a WCF answer, which I would have assumed if I had seen the question tagged with WCF even if I didn't mention it in the question or title.   : It looks probable that the answer (if there is one) is related to AOP, but the questioner does not require it. What's the action of banging two beers together called? When beers in cans or bottles are banged together in rejoice of the company you're with, what is the action called? Why can rm remove read-only files? If I create a file and then change its permissions to 444 (read-only), how come rm can remove it?  If I do this:  echo test &gt; test.txt chmod 444 test.txt rm test.txt   ...rm will ask if I want to remove the write-protected file test.txt. I would have expected that rm can not remove such a file and that I would have to do a chmod +w test.txt first. If I do rm -f test.txt then rm will remove the file without even asking, even though it's read-only.   Can anyone clarify? I'm using Ubuntu 12.04/bash. What the #$@&%*! is that called? Is there a name for the use of symbols in place of curse words, for example #$@&amp;%*!? Proof that a sum of Bernoulli rvs has Binomial distribution Let $\{X_n, n \geq 1\}$ be iid Bernoulli random variables with $\Pr(X_1 = 1) = p = 1 - \Pr(X_1 = 0)$ and let $S_n = \sum\limits_{i=1}^n X_i$ be the number of successes in $n$ trials. Show $S_n$ has a binomial distribution by the following methods:  1.) Prove for $n\geq0,$ $1\leq{k}\leq{n+1}$  $$\Pr(S_{n+1} = k) = p\Pr(S_n = k - 1) + q\Pr(S_n = k)\text{ where }q = 1-p.$$  2.) Solve the recursion using generating functions.  For the first method, I think I need to use convolution of $n + 1$ Bernoulli rvs, but I cannot immediately see the progression. For the second method, I don't exactly understand how to proceed either. This is problem 1.2 from Resnick's Adventures in Stochastic Processes. What are Identity Columns? I was reviewing the  for PostgreSQL and I saw that Pg is likely going to get "identity columns" sometime soon.  I found some mention in  but nothing much  is_identity         yes_or_no         Applies to a feature not available in PostgreSQL identity_generation character_data    Applies to a feature not available in PostgreSQL identity_start      character_data    Applies to a feature not available in PostgreSQL identity_increment  character_data    Applies to a feature not available in PostgreSQL identity_maximum    character_data    Applies to a feature not available in PostgreSQL identity_minimum    character_data    Applies to a feature not available in PostgreSQL identity_cycle      yes_or_no         Applies to a feature not available in PostgreSQL    doesn't say much either     An identity column differs from a primary key in that its values are managed by the server and usually cannot be modified. In many cases an identity column is used as a primary key; however, this is not always the case.   But, I don't see anything else on them. How do identity columns work? Do they provide any new functionality or is this just a standard method to create sequences? Any breakdown of the new feature and how it works? Comparing areas in scanned historical land use maps using QGIS? Which is correct?     How much is the two fares?      How much are the two fares? Central limit theorem and the law of large numbers I have a very beginner's question regarding the Central Limit Theorem (CLT):  I am aware that the CLT states that a mean of i.i.d. random variables is approximately normal distributed (for $n \to \infty$, where $n$ is the index of the summands) or the standardized random variable would have a standard normal distribution.  Now the Law of Large Number states roughly speaking that the mean of i.i.d random variables converges (in probability or almost surely) to their expected value.  What I don't understand is: If, as the CLT states, the mean is approximately normally distributed, how then can it also converge to the expected value at the same time?   Convergence would imply for me that with time the probability that the mean takes a value which is not the expected value is almost zero, hence the distribution would not really be a normal but almost zero everywhere except at the expected value.  Any explanation is welcome. I currently have the Geometry Dash lite version and I want to buy the full version. Can I login with my account of Geometry Dash lite and keep everything I already had? It would suck if I had to redo everything. How do I read from /proc/$pid/mem under Linux? The  tells me that /proc/$pid/mem “can be used to access the pages of a process's memory”. But a straightforward attempt to use it only gives me  $ cat /proc/$$/mem /proc/self/mem cat: /proc/3065/mem: No such process cat: /proc/self/mem: Input/output error   Why isn't cat able to print its own memory (/proc/self/mem)? And what is this strange “no such process” error when I try to print the shell's memory (/proc/$$/mem, obviously the process exists)? How can I read from /proc/$pid/mem, then? top output: cpu usage > 100%