Let $X=(X_1,X_2,...,X_n)$ be an independent multivariate random variable, i.e. $$ F_X(x_1, x_2,...,x_n)=\prod_{i=1}^{n}F_{X_i}(x_i) $$ This implies that if we take 2 components $X_i, X_j$, these are independent as well: $$ F_{(X_i, X_j)}(x,y)=F_{X_i}(x) \cdot F_{X_j}(y) $$ I need to prove that the reverse statement is not true, i.e this is false: $$ F_{(X_i, X_j)}(x,y)=F_{X_i}(x) \cdot F_{X_j}(y) \implies F_X(x_1, x_2,...,x_n)=\prod_{i=1}^{n}F_{X_i}(x_i) $$  A theoretical proof or a counter-example are both acceptable. Thanks, András
Show by a counterexample that for a family $(X_i)_{i\in I}$ of random variables the independence of all pairs $(X_i,X_j)$ with $i,j\in I, i\neq j$ does not imply the independence of the family (It is enough to consider $\mbox{card}(I)=3$.).      Hello, I am searching for a simple example. The professor gave the hint that one might consider random variables which only can take two different values, but I do not know exactly what is meant.  By the way:  We call the family $(X_i)_{i\in I}$ of random variables independent, if $(\sigma(X_i))_{i\in I}$ is independent, whereat we call a family $(\mathcal{E}_i)_{i\in I}$ with $\mathcal{E}_i\subset\mathcal{A}$ independent, if for any finite subset $I_0\subset I$ and any choice $E_i\in\mathcal{E}_i, i\in I_0$, it is $$ \mathbb{P}(\bigcap_{i\in I_0}E_i)=\prod_{i\in I_0}\mathbb{P}(E_i). $$
Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$ Problem:   If $f(x)$ is continous at $x=0$, and $\lim\limits_{x\to 0} \dfrac{f(ax)-f(x)}{x}=b$, $a, b$ are constants and $|a|&gt;1$, prove that $f'(0)$ exists and $f'(0)=\dfrac{b}{a-1}$.   This approach is definitely wrong:    \begin{align} b&amp;=\lim_{x\to 0} \frac{f(ax)-f(x)}{x}\\ &amp;=\lim_{x\to 0} \frac{f(ax)-f(0)-(f(x)-f(0))}{x}\\ &amp;=af'(0)-f'(0)\\ &amp;=(a-1)f'(0) \end{align}  I will show you a case why this approach is wrong:     \[f(x)= \begin{cases} 1,&amp;x\neq0\\ 0,&amp;x=0 \end{cases}\]   $\lim_{x\to0}\dfrac{f(3x)-f(x)}{x}=\lim_{x\to0} \dfrac{1-1}{x}=0$   but $\lim_{x\to0}\dfrac{f(3x)}{x}=\infty$,$\lim_{x\to0}\dfrac{f(x)}{x}=\infty$   Does anyone know how to prove it? Thanks in advance! NullPointerException accessing views in onCreate() I am trying to get Juju working on Windows 8 but I am running in to some errors when trying to get juju to see my ssh keys:  C:\Users\username&gt; juju bootstrap error: error parsing environment "azure":     read C:\Users\user\SkyDrive\Documents\Azure\ssh\: The handle is invalid.   I've added a public key I generated with putty to the directory above called azure  My environments.yaml file has this in it:  authorized-keys-path: C:\Users\user\SkyDrive\Documents\Azure\ssh\   Any ideas? pgfmathanglebetweenpoints always returns 90 DOM event precedence What order of precedence are events handled in JavaScript?  Here are the events in alphabetical order...   onabort - Loading of an image is interrupted onblur - An element loses focus onchange - The user changes the content of a field onclick - Mouse clicks an object ondblclick - Mouse double-clicks an object onerror - An error occurs when loading a document or an image onfocus - An element gets focus onkeydown - A keyboard key is pressed onkeypress - A keyboard key is pressed or held down onkeyup - A keyboard key is  released onload - A page or an image is  finished loading onmousedown - A mouse button is  pressed onmousemove - The mouse is moved onmouseout - The mouse is moved off  an element onmouseover - The mouse is moved  over an element onmouseup - A mouse button is  released onreset - The reset button is  clicked onresize - A window or frame is  resized onselect - Text is selected onsubmit - The submit button is  clicked onunload - The user exits the page   What order are they handled out of the event queue?  The precedence is not first-in-first-out (FIFO) or so I believe. If $p$ and $q = 2p + 1$ are both odd primes, show that $-4$ and $2(-1)^{(1/2)(p-1)}$ are both primitive roots modulo $q$.   I cannot get heads nor tails of how to even start this let alone finish it I am having trouble with the following problem:   $f: N\times N\rightarrow N$ and $f(i,j)=2^{i-1}(2j-1)$. Prove that $f$ is a bijection thus $N\times N$ and $N$ are numerically equivalent.   Work: I am told that Euclid's Lemma will be useful to proving injection and also that for any positive integer $n$, $n$ is a product of prime numbers.   For injection, I set $f(a,b)=f(x,y)\rightarrow 2^{a-1}(2b-1)=2^{x-1}(2y-1)$.  Then I distributed across to get $2^ab-2^{a-1}=2^xy-2^{x-1} \rightarrow 2^ab-2^xy=2^{a-1}-2^{x-1} \rightarrow 2(2^{a-1}b-2^{x-1}y)=2^{a-1}-2^{x-1}$.   From here, I am unsure how to apply Euclid's Lemma to arrive at the injection result.  I am even more unsure how to prove surjection. Proper submodules of Noetherian modules are intersections of indecomposable modules If $M$ is Noetherian, any submodule $N \neq M$ can be written as a finite intersection of intersection-indecomposable ones.   I just don't know how to begin with the proof of this proposition.   Very kind of you for reading or answering. What's the Best Way to Shuffle an NSMutableArray? If you have an NSMutableArray, how do you shuffle the elements randomly?  (I have my own answer for this, which is posted below, but I'm new to Cocoa and I'm interested to know if there is a better way.)    Update: As noted by @Mukesh, as of iOS 10+ and macOS 10.12+, there is an -[NSMutableArray shuffledArray] method that can be used to shuffle. See  for details. (But note that this creates a new array, rather than shuffling the elements in place.) Are Complex Substitutions Legal in Integration? This question has been irritating me for awhile so I thought I'd ask here.  Are complex substitutions in integration okay?  Can the following substitution used to evaluate the Fresnel integrals:  $$\int_{0}^{\infty} \sin x^2\, dx=\operatorname {Im}\left( \int_0^\infty\cos x^2\, dx + i\int_0^\infty\sin x^2\, dx\right)=\operatorname {Im}\left(\int_0^\infty \exp(ix^2)\, dx\right)$$  Letting $ix^2=-z^2 \implies x=\pm\sqrt{iz^2}=\pm \sqrt{i}z \implies z=\pm \sqrt{-i} x \implies dx = \pm\sqrt{i}\, dz$  Thus the integral becomes  $$\operatorname {Im}\left(\pm \sqrt{i}\int_0^{\pm\sqrt{-i}\infty} \exp(z^2)\, dz\right)$$  This step requires some justification, and I am hoping someone can help me justify this step as well: $$\pm \sqrt{i}\int_0^{\pm\sqrt{-i}\infty} \exp(z^2)\, dz=\pm\sqrt{i}\int^\infty_0\exp(z^2)\, dz=\pm\sqrt{i}\left(\frac{\sqrt{\pi}}{2}\right)$$  Thus  $$\operatorname {Im}\left(\int_0^\infty \exp(ix^2)\, dx\right)=\operatorname {Im}\left(\pm\frac{\sqrt{i\pi}}{2}\right)=\operatorname {Im}\left(\pm\frac{(1+i)\sqrt{\pi}}{2\sqrt{2}}\right)=\pm\frac{1}{2}\sqrt{\frac{\pi}{2}}$$  We find that the correct answer is the positive part (simply prove the integral is positive, perhaps by showing the integral can be written as an alternating sum of integrals).  Can someone help justify this substitution?  Is this legal? If $A,B,C$ are matrices I am thinking how to show that $$ A(B + C) = AB + AC$$  Is possible to show without sums like $\sum_i a_i, ..., \sum_j b_j$? It seems if I do the proof with many indexes then is tedious and I don't learn much from it. Can the Ubuntu logo be used for a commercial website? Is there an Ubuntu logo that can be used on a commercial website to indicate that Ubuntu can be installed for customers? Basic voltage divider issue in my amateur circuit XBee Once again I need your help in my project. I have a 4,5v powered circuit. I put a voltage divider with 2 resistors in series to reduce my initial voltage of 4.5 V to 3.4 V (to feed an XBee module).  The issue is that if I do not connect the xbee, the voltage divider reduces the input voltage to the desired voltage(3.4 V), but when I connect the XBee module, the resulting voltage divider voltage drops to a value very low (0.85 V), making the XBee malfunction. Surely it is a basic mistake, I am asking something that is very clear, but I'm newbies.   Basic Circuit:  Vin(4.5 V)-----R2(10 kΩ)---------R1(33 kΩ)--------GROUND                        |                        |                  Vout desired (3.4 V)                        |                        |                       XBee My professor was lecturing today and he made this statement which I was unable to verify.  (I worded it nicer)  There is no map which is both smooth and onto from $S^1$ to $S^1$$\times$ $S^1$.  When he said this he included the original "one can clearly see" statement which is usually when the most people seem to be confused in his class.  Thus, I was hesitant to ask why.  So, please no one lecture me on the fact that I should have asked anyway.  If it is really that obvious, I would rather look stupid here than in class :) How do the bonus perks from the Advanced Warfare Center work? GNU screen: Running commands in different window when screen is started How does the Schengen 90/180 rule work? Python "if X == Y and Z" syntax copy files over ssh Out of five needed members for my doctoral thesis committee, four are fixed. For the fifth member, I approached Mr.A. He has no real connection to my research or topic but I contacted him because he was just as good a fit as anybody else in the department to be the fifth member. I have never worked with him or taken any classes but we did talk a lot in the first few years where he would just ask about my progress by the copy machine, etc. We don't even have a "rapport". The relationship is absolutely neutral, nothing positive/friendly, and nothing negative. I asked him to be on my committee and he said yes. I said I'll get back to you with dates, etc. He is a senior tenured professor.  Then about a month later, I was looking at the CV of some of the new hired (junior) faculty in the department Mr.B (this is his first year) and I see that we have a collaborator in common. Mr.B has published multiple papers with him and Mr.B is working with the same code and the code managing team that we are working with. He is definitely a good person to have on my committee. I finally see Mr.B after a few more months (because of busy schedule/travel/etc.) and he agrees to be on my committee.  The question is, how can I tell Mr.A, as tactfully as possible, after about four months of being quiet that he is basically rejected from my committee because I didn't know about Mr.B but now that I know, Mr.B is a much better match than he is? That I don't need him but I do appreciate his offer? I don't want to antagonize Mr.A or burn any bridges. Should this be in email or in person? Should I explain or just keep it terse? Should I keep it vague that I don't need him and thanks for his time? Or should I tell him about Mr.B and why he is a better match?  FYI, this was before anything official/paperwork was done so there is no problem with that.