I know statistical test usually requires &quot;enough&quot; samples to estimate the uncertainties. But what if I have only one sample in each group, and also I &quot;know&quot; which distribution this one sample comes from, is it reasonable to construct a statistical test? To be more specific, Let's say we have a case-control study and we want to know if there is some significant improvement in &quot;case&quot; than in &quot;control&quot;. The data we have is only one sample, $x$ for the case group, and $y$ for the control group, so $x$ and $y$ are both scalars. The other information we also have is that $X\sim exp(\lambda)$ and $Y\sim exp(\mu)$, where $\lambda$ and $\mu$ are both some known parameters. My question is to test my hypothesis that &quot;case is greater than control&quot;, can I just construct a test statistic $T = X-Y$? And I will be able to find the distribution of $T$ by this . Does this make sense? My confusion mainly comes from how to write up $H_0$ and $H_1$. It seems like I am testing $H_0:x=y$ but obviously this looks very weird... Because usually we will test $H_0: \lambda = \mu$...
I am wondering what we can say, if anything, about the population mean, $\mu$ when all I have is one measurement, $y_1$ (sample size of 1).  Obviously, we'd love to have more measurements, but we can't get them.  It seems to me that since the sample mean, $\bar{y}$, is trivially equal to $y_1$, then $E[\bar{y}]=E[y_1]=\mu$.  However, with a sample size of 1, the sample variance is undefined, and thus our confidence in using $\bar{y}$ as an estimator of $\mu$ is also undefined, correct?  Would there be any way to constrain our estimate of $\mu$ at all?
Primary ideals in Noetherian rings Should Luke's lightsaber have given away his plan? During Luke's confrontation of Kylo Ren toward the end of The Last Jedi we see that       the lightsaber Luke uses a blue lightsaber (which I thought was his original one but that really doesn't matter), not the green one of his own construction.   Given that Kylo Ren  and Rey has just     fought over and broken the blue lightsaber   Why did Kylo Ren not notice this "inconsistency" and realize that      Luke's physical body was not there? How is consensus determined on Meta sites? On our main sites &quot;the best answer&quot; is  as the superposition of the highest voted answer and the accepted answer.  Truth be told, the system is rather insistent on allowing multiple &quot;right answers&quot; and that's one of its strengths: you can be the judge of who you believe. But that doesn't always translate well on the per-site metas, which serve as a sort of policy-creation body.  People who are active on meta help decide how policies are enforced on the main site.  Moderators who have ♦-powers tend to take mandates on meta very seriously.  , accepted answers to meta-questions are no different than any other answer.  Does that leave just votes to determine consensus? Closely related: Not everyone participates on meta-sites.  It can be hard to tell if enough people have looked and voted on questions for real consensus to be declared.  Size of the site determines how many people could be voting on questions.  What counts as a quorum? Finally, sometimes people change their minds about a meta-question, but, , it's not really possible to change one's mind without a pointless edit to someone else's answer.  When should meta-questions be revisited?  On graduated sites, ♦ moderators are elected, which makes them representatives of the active participants, but on beta sites, we are appointed.  It's doubly pressing for beta sites to determine &quot;the will of the people&quot; since part of the point of a beta is to get a working policy framework in place before electing permanent moderators. I'm interested in figuring out how sites already answer these questions and what might be done network-wide to make discovering consensus a little bit easier.  N.B.: Obviously this assumes that consensus matters.  It does to me when I'm making my decisions and I suspect it matters to others.  But that isn't to say that it always makes a difference.  I'm really not interested in answers that hope to use this question as a grindstone for all the axes stored up against certain people who have made unpopular decisions and have the power to make those decisions stick.  Feel free to ask your own question(s).  (Thank you.) Why didn't Escobar's hippos introduced in a single event die out due to inbreeding Today I read a  about how Pablo Escobar had once imported 4 hippos (1 male, 3 female) into his estate in Colombia for his private zoo. After his downfall, while other species were shipped out, hippos were considered too big to move and expected to not survive. However, to the surprise of all the hippos are thriving and are so numerous that there have been calls to cull them. From the report:  Numbers are projected to only get bigger. [Colombian biologist Nataly] Castelblanco and her peers say the population will reach over 1,400 specimens as early as 2034 without a cull - all of them descended from the original group of a male and three females. In the study, they envisaged an ideal scenario in which 30 animals need to be culled or castrated every year to stop that happening.  My understanding is that since there was only 1 male, the gene pool would be limited and lead to lot of inbreeding in the descendants. This would cause population to not explode because some individuals would be unfit to survive. Why has this not happened in case of hippos? Is it because there are 3 females (probably unrelated to each other) which keeps gene pool large enough? Or can mutations explain this phenomenon? Would results have been different if originally 2 females had been moved and only 1 retained? EDIT - I have just found that  is maybe less than 100 individuals which though big is not massive. EDIT2 - I have edited the question title to keep focus on the hippos although a general answer would be welcome. pfsense more than 2 isp provider is it possible to bridge more than 2 isp provider using pfsense? what router or device can bridge 2 or more isp providers in a router? i am planning to bridge at lease 5 isp providers to create a very good internet connection, what can i do to link all 5 providers in 1 router? Why is the well ordering principle counter-intuitive? I read  that while 'The Axiom of Choice agrees with the intuition of most mathematicians; the Well Ordering Principle is contrary to the intuition of most mathematicians'. I don't understand why this is so. According to Wikipedia, 'the well-ordering principle states that every non-empty set of positive integers contains a least element'. This is quite a straightforward definition. Is it the way you derive the equivalence of the Axiom of Choice and well ordering principle that makes it counter-intuitive or the definition of the least element? Sequence Limit Problem: If $0 \leq x_{m+n} \leq x_n + x_m$ then limit of $x_n/n$ exists How can I show or hide boot messages when Ubuntu starts? "Made out of", "made of" and "made with" I understand the phrases made of and made from from the questions and answers provided  and .  Yet I came across three more phrases: made out of, made of, and made with which I think are related to the subject in question.  Would you please help me understand how to differentiate these phrases too? And how is their usage? Solution of trigonometric equation $0 = x\cos(x)+2$ Is there an GUI Designer for python? Is there a GUI IDE for programming python just like java has netbeans?  I wanted one where we could drag &amp; drop test boxes &amp; labels just as we do in swing components using netbeans or eclipse.  Or is there an add-on present in either netbeans or eclipse itself? Deriving of formula for finding the length of median In the below image    $AD$ is the median of $\triangle ABC$  We know that $m_A = \frac 1 2 \sqrt{2b^2 + 2c^2 - a^2} $    But can someone tell me how it's derived !! I am just unable to think of it !!   But I assume it muste be linked with the Appolonius Theorem !! What is the correct JSON content type? I've been messing around with  for some time, just pushing it out as text and it hasn't hurt anybody (that I know of), but I'd like to start doing things properly.  I have seen so many purported "standards" for the JSON content type:  application/json application/x-javascript text/javascript text/x-javascript text/x-json   But which one is correct, or best? I gather that there are security and browser support issues varying between them.  I know there's a similar question, , but I'd like a slightly more targeted answer. How much longer will my 10 year old frame last? Sort arrays of primitive types in descending order The following example must be compiled with XeLaTeX to reproduce the issue.  % arara: xelatex \documentclass{article} \usepackage{fontawesome} \begin{document} Text Text \faMobilePhone\ Text Text \end{document}   Normally I am using TeXworks to compile and view the output. With the internal viewer I am getting the desired result. The following image demonstrates it:    However if I open the pdf with Preview (viewer of Mac) the scaling of the icon fails.     If I open the pdf file with Adobe Acrobat Reader the icon is still scaled correctly. If I print the page my printer is also not able to scale the icons correctly.   If I use LuaLaTeX everything seems to work correctly.   Can you reproduce the issue and do you know a solution? How can I give a script its own icon in the Unity launcher? With Ubuntu 16.04, if you create a launcher (desktop file in ~/.local/share/applications) with a custom icon for a terminal application (so with Terminal=true) then when you start it, it will spawn a new standard terminal icon, and your custom icon will pulse and disappear within a few seconds.  In 14.04, it just worked as expected. (it would not start a new standard terminal icon).   Any idea what to do to change this behaviour? I have a few terminal apps that I want to launch from unity and the new behaviour is problematic (I lose trace of which is which since they all end up with the standard terminal icon)... Search and download audio-only from youtube Is there a way to search and then download from YouTube, with the situation that I only need to download the audio only, not the video. What are the differences between Factor Analysis and Principal Component Analysis? How to prove that the sum and product of two algebraic numbers is algebraic? Suppose $E/F$ is a field extension and $\alpha, \beta \in E$ are algebraic over $F$. Then it is not too hard to see that when $\alpha$ is nonzero, $1/\alpha$ is also algebraic. If $a_0 + a_1\alpha + \cdots + a_n \alpha^n = 0$, then dividing by $\alpha^{n}$ gives $$a_0\frac{1}{\alpha^n} + a_1\frac{1}{\alpha^{n-1}} + \cdots + a_n = 0.$$     Is there a similar elementary way to show that $\alpha + \beta$ and $\alpha \beta$ are also algebraic (i.e. finding an explicit formula for a polynomial that has $\alpha + \beta$ or $\alpha\beta$ as its root)?    The only proof I know for this fact is the one where you show that $F(\alpha, \beta) / F$ is a finite field extension and thus an algebraic extension.