Why is independence of observations an assumption in logistic regression I am currently learning about the assumptions of logistic regression and am having a hard time wrapping my head around why independence of observations is necessary for this test. Any guidance would be appreciated.
Is there i.i.d. assumption on logistic regression? Is there i.i.d. assumption on the response variable of logistic regression?  For example, suppose we have $1000$ data points. It seems the response $Y_i$ is coming from a Bernoulli distribution with $p_i=\text{logit}^{-1}(\beta_0+\beta_1 x_i)$. Therefore, we should have $1000$ Bernoulli distributions, with different parameter $p$.  So, they are "independent", but are not "identical".  Am I right?    PS. I learned logistic regression from "machine learning" literature, where we optimize the objective function and check if it is good in testing data, without talking too much about assumptions.   My question started with this post  Where I try to learn more on statistical assumptions.
Prove that $\sum_{k=0}^n a_k x^k = 0$ has at least $1$ real root if $\sum_{k=0}^n \frac{a_k}{k+1} = 0$ Knowing that $$ \frac{a_0}{1} + \frac{a_1}{2} + \frac{a_2}{3} +\cdots + \frac{a_n}{n+1} =0$$ Prove that  $$ a_0 + a_1x + a_2x^2 + \cdots + a_nx^n = 0$$ has at least one real solution.  I suspect that it's proven with the intermediate value theorem. But can't find two numbers that satisfy it. While studying the Fibonacci sequence I encountered this problem in the handout, and I can not understand how to do it.     Show that if the Fibonacci sequence has a term divisible by a natural number $m$, then there are infinitely many such terms. If I have flagged a post, show me a "flags" link in my profile What is rank deficiency, and how to deal with it? Fitting a logistic regression using  ends with   Error in mer_finalize(ans) : Downdated X'X is not positive definite.    A likely cause of this error is apparently rank deficiency. What is rank deficiency, and how should I address it? There's a theorem of Kaplansky that states that if an element $u$ of a ring has more than one right inverse, then it in fact has infinitely many. I could prove this by assuming $v$ is a right inverse, and then showing that the elements $v+(1-vu)u^n$ are right inverses for all $n$ and distinct.  To see they're distinct, I suppose $v+(1-vu)u^n=v+(1-vu)u^m$ for distinct $n$ and $m$. I suppose $n&gt;m$. Since $u$ is cancellable on the right, this implies $(1-vu)u^{n-m}=1-vu$. Then $(1-vu)u^{n-m-1}u+vu=((1-vu)u^{n-m-1}+v)u=1$, so $u$ has a left inverse, but then $u$ would be a unit, and hence have only one right inverse.  Does the same theorem hold in monoids, or is there some counterexample? Does the use of words such as 'fail', 'wrong', 'incorrect' and 'invalid' have a sufficiently negative impact on a users frame of mind to warrant the prohibition of their inclusion in the error messages generated by typical web forms?  i.e. Should we rephrase examples like these to just use positive words?  'date entered incorrectly'  or  'failed to complete registration, please see below for invalid field entries'  (PS, the examples are there to show use of the words in an error message, not to serve as examples of error messages I would use or things that need correcting in answers, I'm not looking for advice on how to write form error messages, but for information about the effect of certain words on a users frame of mind) Undefined Control Sequence in \multido or \pspicture? Division by $0$ I came up some definitions I have sort of difficulty to distinguish. In parentheses are my questions.   $\dfrac {x}{0}$ is Impossible (  If it's impossible it can't have neither infinite solutions or even one. Nevertheless, both $1.$ and $2.$ are divided by zero, but only $2. $ has infinite solutions so as $1.$ has none solution, how and why ?) $\dfrac {0}{0}$ is Undefined and has infinite solutions. (How come one be Undefined and yet has infinite solutions ?) $\dfrac {0}{x}$ and $x \ne 0$, it's okay for me, no problem, but if someone else wants to add something about it, feel free to do it. Cant install via Ubuntu Software- "apt transaction returned result exit-failed" I'm using 17.04 ( Clean Install ), and as far as I remember, every try to install any sofware- it fails with that error code.  I tried sudo apt-get update &amp;&amp; apt-get upgrade and problem persists.  any idea why?   edit: Using software Updater gave this result :   edit : output of update  guy@LG-LAPTOP:~$ sudo apt-get update &amp;&amp; apt-get upgrade [sudo] password for guy:  Ign:1 http://il.archive.ubuntu.com/ubuntu zesty InRelease Ign:2 http://il.archive.ubuntu.com/ubuntu zesty-updates InRelease               Ign:3 http://il.archive.ubuntu.com/ubuntu zesty-backports InRelease             Err:4 http://il.archive.ubuntu.com/ubuntu zesty Release                           503  Service Unavailable Get:5 http://security.ubuntu.com/ubuntu zesty-security InRelease [89.2 kB]      Err:6 http://il.archive.ubuntu.com/ubuntu zesty-updates Release                   503  Service Unavailable Get:7 http://archive.canonical.com/ubuntu zesty InRelease [10.2 kB]             Err:8 http://il.archive.ubuntu.com/ubuntu zesty-backports Release                 503  Service Unavailable Get:9 http://archive.canonical.com/ubuntu zesty/partner Sources [1,416 B]       Get:10 http://archive.canonical.com/ubuntu zesty/partner amd64 Packages [1,616 B] Get:11 http://archive.canonical.com/ubuntu zesty/partner i386 Packages [1,612 B] Get:12 http://archive.canonical.com/ubuntu zesty/partner Translation-en [708 B] Get:13 http://security.ubuntu.com/ubuntu zesty-security/main amd64 DEP-11 Metadata [11.7 kB] Get:14 http://security.ubuntu.com/ubuntu zesty-security/main DEP-11 64x64 Icons [9,219 B] Get:15 http://security.ubuntu.com/ubuntu zesty-security/universe amd64 DEP-11 Metadata [14.4 kB] Get:16 http://security.ubuntu.com/ubuntu zesty-security/universe DEP-11 64x64 Icons [31.0 kB] Reading package lists... Done                                          E: The repository 'http://il.archive.ubuntu.com/ubuntu zesty Release' does no longer have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details. E: The repository 'http://il.archive.ubuntu.com/ubuntu zesty-updates Release' does no longer have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details. E: The repository 'http://il.archive.ubuntu.com/ubuntu zesty-backports Release' does no longer have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details. guy@LG-LAPTOP:~$    output of apt get  guy@LG-LAPTOP:~$ sudo apt-get install putty Reading package lists... Done Building dependency tree        Reading state information... Done The following additional packages will be installed:   putty-tools Suggested packages:   putty-doc The following NEW packages will be installed:   putty putty-tools 0 upgraded, 2 newly installed, 0 to remove and 0 not upgraded. Need to get 661 kB of archives. After this operation, 2,713 kB of additional disk space will be used. Do you want to continue? [Y/n]  Err:1 http://il.archive.ubuntu.com/ubuntu zesty/universe amd64 putty-tools amd64 0.67-3   503  Service Unavailable Err:2 http://il.archive.ubuntu.com/ubuntu zesty/universe amd64 putty amd64 0.67-3   503  Service Unavailable E: Failed to fetch http://il.archive.ubuntu.com/ubuntu/pool/universe/p/putty/putty-tools_0.67-3_amd64.deb  503  Service Unavailable E: Failed to fetch http://il.archive.ubuntu.com/ubuntu/pool/universe/p/putty/putty_0.67-3_amd64.deb  503  Service Unavailable E: Unable to fetch some archives, maybe run apt-get update or try with --fix-missing? guy@LG-LAPTOP:~$ $\sigma$-algebra vs. $\sigma$-field: is there any difference? The subject says it all: is there any difference between the two concepts of $\sigma$-algebra and $\sigma$-field? In probability theory, they seem to be used more or less interchangeably. If there is no difference, is there any historical reason why some people/schools use the term $\sigma$-algebra, while others use the term $\sigma-$field? Proof that the relation $5 \mid (a + 4b)$ is symmetric and transitive Let $A,B$ be two sets. The  states that if there is an injection $f\colon A\to B$ and an injection $g\colon B\to A$, then there exists a bijection $h\colon A\to B$.  I was wondering whether the following statements are true (maybe by using the AC if necessary):   Suppose $f \colon A\to B$ and $g\colon B\to A$ are both surjective, does this imply that there is a bijection between $A$ and $B$. Suppose either $f\colon A\to B$ or $g\colon  A\to B$ is surjective and the other one injective, does this imply that there is a bijection between $A$ and $B$. How can you reduce ambient noise from servers? I have several AppExchange apps installed that are failing tests. What is the recommended approach to resolving this? Are tests run when a new package is installed? Is there a way to determine what is actually failing besides the name of the method and something like "expected true, actually false"? The test results rarely show enough information to be useful.   What I'm doing atm, is checking for updates for the packages, and then writing the company with the failing test results.   Any feedback would be appreciated. What is this shell/Bash syntax: someVariable=someValue someCommand Python super() raises TypeError In Python 2.5, the following code raises a TypeError:  &gt;&gt;&gt; class X:       def a(self):         print "a"  &gt;&gt;&gt; class Y(X):       def a(self):         super(Y,self).a()         print "b"  &gt;&gt;&gt; c = Y() &gt;&gt;&gt; c.a() Traceback (most recent call last):   File "&lt;stdin&gt;", line 1, in &lt;module&gt;   File "&lt;stdin&gt;", line 3, in a TypeError: super() argument 1 must be type, not classobj   If I replace the class X with class X(object), it will work. What's the explanation for this? How could I get the terminal to be launched once I have logged in, so that typing commands is much simpler? Why is there something instead of nothing? A simple but fundamental question.    The "something" means the whole Universe (known and unknown), it could be represented as the reality version of the set of all sets, which is itself . It includes all the  and such.  A better version of the whole Universe could be described with a .  The "nothing" means an absence of the whole Universe, not a particular void or emptiness in the whole Universe but a nonexistent whole Universe. If nonempty, nonsingleton $Y$ is a proper convex subset of a simply ordered set $X$, then $Y$ is ray or interval? This is from Exercise 7 in p. 92 in Munkres's Topology.  Except for the trivial cases such as $Y$ is empty set or singleton, it seems if $Y$ is convex in an simply ordered set $X$ then $Y$ is interval or ray.  But I cannot start my proof because I cannot use $\sup$ or $\max$ functions.  What should I do? undelete big files - mission impossible? I've accidentaly deleted outlook.pst (6.7GB) file, while there was only 400MB free space left on primary NTFS partition (winxp).  I've tried several recovery tools to get this file back. "Ontrack Easy Recovery Pro" found 0 pst files (complete scan mode), while "Recover My Files" in sector scan mode found 5 pst's, but 4 of them of sizes from 3 to 28 KB, while the 5th one - 1Gb.  I've managed to succesfuly recover 1Gb pst file, which was 1 year old copy (the one used after the latest windows reinstall).  Now, I'm frustrated and confused   Why 1 year old file was succesfuly recovered if there were only 400MB left on primary partition? Where's 6.7GB file gone? I did some reading (), and it seems that there's almost no probability to retrieve the file I'm looking for, but wait - none of recovery tools i've used found zero-sized pst file, moreover - if due to fragmentation a file might be corrupted - we could use scanpst.exe to fix some errors and survive with 10 or 100 emails missing - whatever. Could you please recommend some more sophisticated recovery tools for this particular task?   Appretiate your help - thanks in advance