Orthogonal Projection onto the Affine set $ \left\{ x \mid A x = b \right\} $
Orthogonal Projection of $ z $ onto the Affine set $ \left\{ x \mid A x = b \right\} $
Why does a 95% Confidence Interval (CI) not imply a 95% chance of containing the mean? It seems that through various related questions here, there is consensus that the "95%" part of what we call a "95% confidence interval" refers to the fact that if we were to exactly replicate our sampling and CI-computation procedures  many times, 95% of thusly computed CIs would contain the population mean. It also seems to be the consensus that this definition does not permit one to conclude from a single 95%CI that there is a 95% chance that the mean falls somewhere within the CI. However, I don't understand how the former doesn't imply the latter insofar as, having imagined many CIs 95% of which contain the population mean, shouldn't our uncertainty (with regards to whether our actually-computed CI contains the population mean or not) force us to use the base-rate of the imagined cases (95%) as our estimate of the probability that our actual case contains the CI?   I've seen posts argue along the lines of "the actually-computed CI either contains the population mean or it doesn't, so its probability is either 1 or 0", but this seems to imply a strange definition of probability that is dependent on unknown states (i.e. a friend flips fair coin, hides the result, and I am disallowed from saying there is a 50% chance that it's heads).  Surely I'm wrong, but I don't see where my logic has gone awry... Prediction interval for binomial random variable What is the formula (approximate or exact) for a prediction interval for a Binomial random variable?  Assume $Y \sim \mathsf{Binom}(n, p)$, and we observe $y$ (drawn from $Y$). The $n$ is known.  Our goal is to obtain a 95% prediction interval for a new draw from $Y$.   The point estimate is $n\hat{p}$, where $\hat{p}=\frac{y}{n}$. A confidence interval for $\hat{p}$ is straightforward, but I cannot find a formula for a prediction interval for $Y$. If we knew $p$ (rather than $\hat{p}$), then a 95% prediction interval just involves finding the quantiles of a binomial. Is there something obvious I'm overlooking? For some reasons, I would like to do an explicit quoting of a string value (becoming a part of constructed SQL query) instead of waiting for implicit quotation performed by cursor.execute method on contents of its second parameter.  By "implicit quotation" I mean:  value = "Unsafe string" query = "SELECT * FROM some_table WHERE some_char_field = %s;" cursor.execute( query, (value,) ) # value will be correctly quoted   I would prefer something like that:  value = "Unsafe string" query = "SELECT * FROM some_table WHERE some_char_field = %s;" % \     READY_TO_USE_QUOTING_FUNCTION(value) cursor.execute( query ) # value will be correctly quoted, too   Is such low level READY_TO_USE_QUOTING_FUNCTION expected by Python DB API specification (I couldn't find such functionality in  document). If not, maybe Psycopg2 provides such function? If not, maybe Django provides such function? I would prefer not to write such function myself... Mathematical expression to form a vector from diagonal elements I would like to know is there any way to express mathematically (using matrices multiplication, addition, etc.) a vector which is formed from the diagonal elements of a matrix. For example, I have a matrix called $\mathbf{M}$ and I want to create a vector  $\mathbf{v}$ such that $\mathbf{v}= \text{diagonal elements of } \mathbf{M} $  Any matrix algebra or operation do that ? Thank you. Query standard input of lightning-input I want to query standard HTML textarea which is wrapped by lightning-textarea. How can I get it?  &lt;lightning-textarea name=&quot;secret_field&quot; label=&quot;Secret&quot; value={secretFieldValue}&gt;&lt;/lightning-textarea&gt; Is there a list of error codes for Minecraft? Similar to  on the bugs in Minecraft, I would like a complete list of what error codes exist in Minecraft and how they occur, as well as how they can be resolved. How to prove that $L=${$a^p$: p is prime} isn't regular? I tried using pumping lemma or finding infinite equivalence classes, but I didn't succeed.  It's clear to me that there is no automata that accepts this language, but I just can't formally prove that this language isn't regular.  How can you prove it? It's using Euclidian inverse. Can you show it step by step?   $$N=7, q=11, a = 3+2X^2 -3X^4 +X^6$$  the inverse of $a \pmod {11}$ is  $$A=-2+4X+2X^2 +4X^3 -4X^4 +2X^5 -2X^6$$  How do we compute $A$? How to delete a file from Time Machine backups in Big Sur? First time this happens to me.  When I try to edit , the title part is immediately erased:   And when I click cancel, the title becomes invisible:     Actually, the revision history is incomplete:   It shows that the question was asked after the comments:   That's highly likely a result of an edition of the history by a moderator.    Google Chrome 62.0.3202.94 (Official Build) (64-bit) on macOS 10.13.2 (17C88). Are there any massless (zero invariant mass) particles carrying electric charge?  If not, why not? Do we expect to see any or are they a theoretical impossibility? How would you cut a shape out of an object using another object? I can see that this works for any integer $n$, but I can't figure out why this works, or why the number $42$ has this property. Cell identification by Body Cells How do the B-cells and T-cells recognize or distinguish a body cell from a foreign particle?  I suspect something to do with this causes Auto Immune Disorders. I'm trying to run down a memory leak in a windows forms application.  I'm looking now at a form which contains several embedded forms.  What worries me is that the child forms, in their constructor, take a reference to the parent form, and keep it in a private member field.  So it seems to me that come garbage-collection time:  Parent has a reference to the child form, via the controls collection (child form is embedded there).  Child form is not GC'd.  Child form has a reference to the parent form, via the private member field.  Parent form is not GC'd.   Is this an accurate understanding of how the garbage collector will evaluate the situation?  Any way to 'prove' it for testing purposes? Modify TikZ coil decoration Can I modify the existing tikz coil decoration, so that the downstrokes are not drawn in a certain interval, to produce a picture like this?    Edit: Previous attempts, code and MWE has been moved to or superseded by my own answer. Installed 14.04 on a non-PAE (Pentium-M), but now I can't update ("This kernel does not support a non-PAE CPU.") Here is what I'm looking at. This isn't my computer. I repair PC's and I had an end user with this low powered laptop running XP so I decided to let him try out lubuntu. The install went smoothly with the non-PAE boot switch, but now I can't update anything. I'd rather not have to format and start over since he's spent time configuring his system. Any tips would be appreciated. I tried sudo apt-get clean, sudo apt-get install -f, sudo apt-get update. No go.  Also his touchpad stopped working, the wireless stopped working, the keyboard works in firefox but not chrome, and in firefox the address bar sometimes turns black and gray for some reason. I was hoping the update would help address some of this stuff.  Here's the message I get trying to update:  Get:1 http://us.archive.ubuntu.com/ubuntu/ trusty-proposed/main linux-image-3.13.0-26-generic i386 3.13.0-26.48 [14.6 MB] Get:2 http://us.archive.ubuntu.com/ubuntu/ trusty-updates/main linux-image-3.13.0-24-generic i386 3.13.0-24.47 [14.6 MB] Fetched 29.2 MB in 2min 19s (208 kB/s)                                          (Reading database ... 169286 files and directories currently installed.) Preparing to unpack .../linux-image-3.13.0-26-generic_3.13.0-26.48_i386.deb ... This kernel does not support a non-PAE CPU. dpkg: error processing archive /var/cache/apt/archives/linux-image-3.13.0-26-generic_3.13.0-26.48_i386.deb (--unpack):  subprocess new pre-installation script returned error exit status 1 Examining /etc/kernel/postrm.d . run-parts: executing /etc/kernel/postrm.d/initramfs-tools 3.13.0-26-generic /boot/vmlinuz-3.13.0-26-generic run-parts: executing /etc/kernel/postrm.d/zz-update-grub 3.13.0-26-generic /boot/vmlinuz-3.13.0-26-generic Preparing to unpack .../linux-image-3.13.0-24-generic_3.13.0-24.47_i386.deb ... This kernel does not support a non-PAE CPU. dpkg: error processing archive /var/cache/apt/archives/linux-image-3.13.0-24-generic_3.13.0-24.47_i386.deb (--unpack):  subprocess new pre-installation script returned error exit status 1 Examining /etc/kernel/postrm.d . run-parts: executing /etc/kernel/postrm.d/initramfs-tools 3.13.0-24-generic /boot/vmlinuz-3.13.0-24-generic run-parts: executing /etc/kernel/postrm.d/zz-update-grub 3.13.0-24-generic /boot/vmlinuz-3.13.0-24-generic Errors were encountered while processing:  /var/cache/apt/archives/linux-image-3.13.0-26-generic_3.13.0-26.48_i386.deb  /var/cache/apt/archives/linux-image-3.13.0-24-generic_3.13.0-24.47_i386.deb E: Sub-process /usr/bin/dpkg returned an error code (1) Let's say that there is a question asked in a bad manner, in a way that the question is missing. For example the user shows the code and doesn't explain very well which is the problem.    Usually someone starts digging in the OP mind posting comments and asking him for details. It may happen that the question gets edited and the OP edits it to a valid question.    Do you usually give the OP time to edit the question or you immediately vote to close it? Random integer in VB.NET I need to generate a random integer between 1 and n (where n is a positive whole number) to use for a unit test. I don't need something overly complicated to ensure true randomness - just an old-fashioned random number.  How would I do that? Possible Duplicate:          I was thinking about making a pacman clone, I know there is a similar question here  , but I know i can't use the original art from the game because it belongs to Namco, so if I design a character that has the shape of the slice circle it will look exactly like pacman, maybe if I use green instead of yellow? Also if the game plays like the original pacman, it is wrong?  I just want to make the game as a personal project and and publish it in my site without getting in trouble