Can I use inferential statistics from a "non-random" sample of 34000? I have 10 years of data on football players individual games which works out to 34000 samples. Can I use this to draw inference about football players in general?  Drawing on 3 requirements of CLT this is what I see:  1) Is it independent?  Yes, this 10 years of data is less than 10% of all football players ever played.  2) Is n>30?  Obviously yes.  3) Random Sample?  This one I'm hung up on. Should I have taken a random sample of the past 10 years? To me it is better to have a lot of observations. Can I use my large sample to draw inference on the larger population or should I be drawing inference on just the past 10 years of players?
Random Sample vs. As Large a Sample as Possible? I'm struggling to differentiate scenarios where you would use random sampling vs. getting as large a sample as possible. For instance, one thing I am trying to test is the notion that being in a "home" arena boosts player performance.  I have collected individual football game data for every game over the past 10 years. I have over 34000 observations of players performance during this time span. Would it have been more prudent for me to take a random sample of say, 1000 players over the past 10 years?  I just don't understand why, in the above scenario, I'd take a random sample of 1000 when I can just take all of the players performances from the past 10 years.
How do I get a Realtek RTL8723BE wireless card to work? How can I install apt packages non-interactively? If I run:  sudo apt-get --yes install postfix    or  sudo bash -c 'yes | apt-get --yes install postfix'   an interactive prompt appears to configure postfix. I need to automate the installation of postfix (I can modify the configuration files after the install).  Is there some magic that lets me install postfix (and other package) without human intervention? how does LEA and several instruction work? How do I remove XFCE from my computer? My computer is overheating because I got too many desktop environments installed like: MATE, Cinnamon and XFCE.  I uninstalled them, but XFCE failed to uninstall. I get this error.   sudo apt-get purge xfce4     Reading package lists... Error!     E: Problem syncing the file - sync (5: Input/output error)     E: The package lists or status file could not be parsed or opened. Windows registry entries for default explorer view Formatting section titles Forgive the stupid simplicity of this question, and please feel free to refer me to an older answer. I'm not quite familiar enough with LaTeX terminology to have found it.  All I want to do is set a different default format for my section titles. Instead of just being bolded and slightly larger than the main text, I'd like them to be of normal weight, the same size as the main text, and all caps.   I'm using the article class. "BAD ARCHIVE MIRROR" error while installing ubuntu server 12.04 I downloaded the server version of Ubuntu from Ubuntu's official site. Created a Bootable USB from the ISO file I downloaded. I connected through Internet using DHCP in the installation process. While I was installing it on my Desktop Computer (pentium D 1 GB RAM) I have this error "Bad Archive Mirror".   I also found 2-3 questions with same problem on forum but I didn't get solution of my problem from them. how to use common modulus attack? I am struck with the following problem:  Let Alice, Bob, Chris and Eve communicate over a public network. They encrypt all messages they send using RSA system. Bob and Chris have the RSA modulus $n_B$ and $n_C$ respectively with $n_B$ = $n_C$ But different public encryption exponents: $e_B \neq e_C$. Suppose $\gcd(e_B, e_C)=1$, and that Alice sends the same secret message to Bob and Chris. Show how Eve can decipher the message. Partitions of $n$ into distinct odd and even parts proof Let $p_\text{odd}(n)$ denote the number of partitions of $n$ into an odd number of parts, and let $p_\text{even}(n)$ denote the number of partitions of $n$ into an even number of parts.  How do I prove that    $|p_\text{even}(n) - p_\text{odd}(n)|$ is equal to the partitions of $n$ into distinct odd parts. Show that the number of partitions of $n$ for which no part appears exactly once is equal to the number of partitions of n for which every part is divisible by 2 or 3. Show that the number of partitions of $n$ for which no part appears more than twice is equal to the number of partitions of $n$ for which no part is divisible by 3. Why is Earth's gravity stronger at the poles? Many sources state that the Earth's gravity is stronger at the poles than the equator for two reasons:   The centrifugal "force" cancels out the gravitational force minimally, more so at the equator than at the poles. The poles are closer to the center due to the equatorial bulge, and thus have a stronger gravitational field.   I understood the first point, but not the second one. Shouldn't the gravitational force at the equator be greater as there is more mass pulling the body perpendicular to the tangent (since there is more mass aligned along this axis)? 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? Switch statement fallthrough in C#? Dealing with "Find out who's going to buy the croissants" On Stack Overflow at the moment, there is a popular question which is currently high in the Hot Questions feed:   So far it has been closed twice (once by the community and a moderator and once by a moderator alone) but re-opened twice by the community. It is clearly not a [so] Stack Overflow question. Reading between the lines, there could be a valid Software Engineering question there, but it would be a very basic one (at the fizz-buzz competency level) and looks more like a populist attempt to farm reputation than a serious question.  Worst of all, even though there are some interesting answers on this question, they are all going to be completely useless to future visitors to the site, since they will be completely undiscoverable. If the question were at least "What is the best fair distribution algorithm?" it would still be off-topic (primarily opinion based), but at least it might help someone once the topic falls off the Hot Questions feed.  Although this has turned into just a bit of fun, and I don't want to kill that, enough is enough. While this question is open, unlocked, and still seen as a valid question, it is setting a bad example for the whole Stack Exchange network.   questions like this, especially on the grandfather Stack Exchange site, devalue the work we are putting in on every other site by making it appear acceptable to re-open populist questions like this after two moderators have classified it as off-topic.  What should we do about this question?   Should it be left open? Power to the people? Closed and locked by a moderator? Migrated to another site? &hellip;or something else? I need to implement the following circuit in LTSpice but I don't know if my schematic is ok. Also, I don't know what  values for I should use for Cinf and R2.  The following constraints are imposed:   VCC= 10V VCM = 1.5V Vgis a sine wave with a frequency of 1 kHz. Its amplitude will be chosen so as to ensure minimal distortion of the output signal (i.e. v0 should also be a sine wave) R1= 15kΩ.    I need to choose a value for R2 and Cinf in order to have a (theoretical)gain value Av=1500.   How can I compute the theoretical expressions of the DC collector currents of bipolar transistors Q1–Q8, considering VBE≈ 0.6V and the theoretical expression of the small-signal voltage gain Av= vo/vg?   The first picture is the schematic and the second is the LTSpice implementation.  What I found is that:   ICq5=ICq6=ICq7=ICq8=(Vcc - Vbe)/R2 ICq1=ICq2 = [ICq7 / 2 ] * [Beta/(Beta+1)] ICq4=iCq2-current of Q5 ICq3=ICq1-current of Q3-current of Q4  Beta in my case is 182.1 because I'm using a BC847A transistor. A=Vo/Vg = Gm_q2 * Beta_q5 * R1Gm_q2=ICq2 / Vt that is approx 40*ICq2   How can I find the value of R2? This is what I have now, an sine like wave for Vg When I run the following query in query editor I get this error.  Error     Unknown error parsing query   Query  SELECT Id, OwnerId,Account.OwnerId FROM Contact where OwnerId = Account.OwnerId limit 100   Does anybody know what the problem is? ASP.NET MVC Default Project Structure and the Scripts Folder When a user without editing privileges suggests an edit to the post, a banner gets displayed until it is reviewed. Can the banner also provide a link to the Suggested edit? For example, I made an edit to a post on Unix&amp;Linux.SE. Can the banner also link to my ?     I know I can access the link through my Profile → Activity → Suggestions. But linking it directly on the face of the post would have immense benefits:  I have been a prolific editor on Ask Ubuntu (my first encounter with Stack Exchange Network) with a lot of Rep derived through Suggestions. I didn't get to know about these suggested edit links until very late (I guess I already had got about 400 edit suggestions approved by the time this came to my knowledge).  This means I didn't know about  when it mattered the most.  I never got to know why my suggestions got rejected because of my ignorance.  This would help reduce the barrier between Reviewers and Editors.  This would also help in case of Anonymous edits. Right now, there is no way an anonymous editor can access it unless the edit gets approved and he comes back to go through the Revision page of that particular post in order to access the Suggested Edit Review link. If the edit gets rejected - it's the end of the road for that anonymous editor. I just made an anonymous edit on one of the SE sites where I don't have an account and already feel like it's the end of the road for me since I just can't access the review link unless it gets approved.    Possibly also expand the banner to mention that they can also access the link to the suggestion via Profile → Activity → Suggestions. How to edit Parameters for an Object after it is created? Who are the users that work for Stack Overflow? I am trying to prove the statement:  If $X\sim\mathcal{N}(0,\sigma_1^2)$ and $Y\sim\mathcal{N}(0,\sigma_2^2)$ are independent random variables, then $\frac{XY}{\sqrt{X^2+Y^2}}$ is also a Normal random variable.  For the special case $\sigma_1=\sigma_2=\sigma$ (say), we have the well-known result that $\frac{XY}{\sqrt{X^2+Y^2}}\sim\mathcal{N}\left(0,\frac{\sigma^2}{4}\right)$ whenever $X$ and $Y$ are independent $\mathcal{N}(0,\sigma^2)$ variables. In fact, it is more generally known that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}}$ are independent $\mathcal{N}\left(0,\frac{\sigma^2}{4}\right)$ variables. A proof of the last result follows by using the transformation $(X,Y)\to(R,\Theta)\to(U,V)$ where $x=r\cos\theta,y=r\sin\theta$ and $u=\frac{r}{2}\sin(2\theta),v=\frac{r}{2}\cos(2\theta)$. Indeed, here $U=\frac{XY}{\sqrt{X^2+Y^2}}$ and $V=\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}}$. I tried to imitate this proof for the problem at hand but it appears to get messy. If I haven't made any error, then for $(u,v)\in\mathbb{R}^2$ I end up with the joint density of $(U,V)$ as $$f_{U,V}(u,v)=\frac{2}{\sigma_1\sigma_2\pi}\exp\left[-\sqrt{u^2+v^2}\left(\frac{\sqrt{u^2+v^2}+v}{\sigma_1^2}+\frac{\sqrt{u^2+v^2}-v}{\sigma_2^2}\right)\right]$$ I have the multiplier $2$ above as the transformation is not one-to-one. So density of $U$ would be given by $\displaystyle \int_{\mathbb{R}}f_{U,V}(u,v)\,\mathrm{d}v$, which isn't readily evaluated. Now I am interested to know if there is a proof where I can only work with $U$ and don't have to consider some $V$ to show that $U$ is Normal. Finding the CDF of $U$ doesn't look so promising to me at the moment. I would also like to do the same for the case $\sigma_1=\sigma_2=\sigma$. That is, if $X$ and $Y$ are independent $\mathcal{N}(0,\sigma^2)$ variables then I wish to show that $Z=\frac{2XY}{\sqrt{X^2+Y^2}}\sim\mathcal{N}(0,\sigma^2)$ without using a change of variables. If somehow I can argue that $Z\stackrel{d}{=}X$, then I'm done. So two questions here, the general case and then the particular case. Related posts on Math.S.E: . . Edit. This problem is in fact due to L. Shepp as I found out in the exercises of An Introduction to Probability Theory and Its Applications (Vol. II) by Feller, alongwith a possible hint:  Surely, $U=\frac{XY}{\sqrt{X^2+Y^2}}=\frac{1}{\sqrt{\frac{1}{X^2}+\frac{1}{Y^2}}}$ and I have the density of $\frac{1}{X^2}$ at hand. Let's see what I could do now. Apart from this, a little help with the integral above is also welcome.