The joint distribution of $(X,Y)$ is given:  $f_{(X,Y)}(x,y)=e^{-(\theta x +\frac{y}{\theta})}$, $0&lt;x,y&lt;\infty$  Find the distribution of $U=XY$ and $V=\frac{X}{Y}$. Also find the conditional pdf of $V$ given $U=u$.  I have transformed $(X,Y)\rightarrow (U,V)$, and found the joint distribution of $(U,V)$ as the following:  $f_{(U,V)}(u,v)=\frac{1}{2v}e^{-(\theta \sqrt {uv}+\frac{1}{\theta}\sqrt {\frac{u}{v}})}$, $0&lt;u,v&lt;\infty$  But I cannot find the marginal pdf of $U$ and $V$ and the conditional pdf of $V$ given $U=u$.  Please anyone help me solve it. Thanks in advance.
Suppose that $(U,V)$ has the following joint pdf: $$f_{U,V}(u,v)=\exp(-\theta u-\theta^{-1}v)$$ , where $u\geq0$, $v\geq0$, $\theta&gt;0$. Define $X=UV$ and $Y=U/V$. Find the marginal pdf of $X$ and $Y$.  With transformation, I got $U=\sqrt{XY}$ and $V=\sqrt{X/Y}$ and the absolute value of Jacob is $1/2Y$. Then, the joint pdf of $(X,Y)$ is  $$f_{X,Y}(x,y)=\exp(-\theta \sqrt{XY}-\theta^{-1}\sqrt{X/Y})1/2Y, X,Y&gt;0.$$  But I am not sure how to get the marginal pdf?  Thank you!
Do I need an ESTA if entering the US from Canada by train As a UK citizen, I will be flying to Vancouver (for which I will need an eTA) and then taking an AMTRAK train to Seattle. I will stay in Seattle for several days before returning to Vancouver and flying back to the UK. Do I need an ESTA for entering the US during this trip? When I have .txt files or .py files, I can Quick Look those files in Finder. I have .jl files that are codes written in the Julia Language. Of course they are text files. They will open in a text editor like Sublime Text. How can I Quick Look these .jl files in Finder? In Portal, we are promised cake and a party at the end of our tests. However;       we soon discover that the cake is a lie.     Then, at the final scene;       there really is a cake (and my companion cube is waiting for me).   What's the meaning of this?  Was GLADOS not deceiving me? Her comments seem to suggest that;       ...she had the intention of killing me. Was the cake for us (player) and not for the character in the game? Limit $\lim_{n\to \infty} \frac{1}{2n} \log{2n\choose n}$ Is there a theorem that says that QFT reduces to QM in a suitable limit? Of course, it should be, as QFT is relativisitc quantum mechanics. But, is there a more manifest one? such as Ehrenfest's theorem which dictates the transition from QM to Classical Mechanics.  I am curious as the basic object in QM is wave function whereas the basic object in QFT is quantized operator, which looks very different, and wondering how the comparison works.  (Of course, I am aware that QFT can calculate corrections for Coulomb potential energy, which may seem to be a transition from QFT to QM, but I want one more manifest than that.) There are two different commands to incorporate another file into the source of some document, \input and \include. When should I use one or the other? What are the differences between them? Are there more things like them to be aware of? Word for "willing to try new and unfamiliar things" Example:  A: Hey, do you like to do bungee jumping? B: Um, OK. Haven't done it before, but let's try it.  I'm not looking for adventurous or something similar. That word should also fit in less action-packed activities:  A: I have a new board game. Do you like to play? B: I don't know this game and, actually, I don't like these varieties of games but I'll play. Maybe it's fun.  First I thought about open-minded but I think this means more willing to listen to people, think about their ideas, arguments, ... and accept them (not necessarily agree). What word describes person who is willing to try activities he doesn't know or hasn't experienced before? I have Java 8 and Ubuntu 18.04. I'm using NetBeans and when tried to make some program, it couldn't be done because JavaFX is missing.  I installed Java JDK with NetBeans from Oracle's site and thought everything is included but obviously I didn't check it well.   What should I do, what should I install? I'm not so good at this and don't know if problem is with Ubuntu or Java. :( A spaceship crew stuck in a damaged spaceship that's losing power Client-Side Rendering/JS Link documentation? Has there been any sort of documentation or anything like that released for SharePoint 2013 Client-Side Rendering/JS Link?  Anthing unofficial? Any dates for when documentation might be released? mathtools: \dddot cause the term to rise Let $\gamma:[0,1] \to \mathbb R^n$ be a smooth closed curve, such that $\gamma(0)=\gamma(1), \gamma'(0)=\gamma'(1), |\gamma'(a)|=1$ and let $B_r=\{x| \exists t, |\gamma(t)-x|&lt;r\}$. How can i show that the volume of $B_r$ is equal to $br^{(n-1)}$ for $r &lt;\varepsilon$ for some $\varepsilon&gt;0$. And how can I find $b$? It is intuitively obvious. But how can one prove it? I have no idea. I'm installing a brand new motion activated security light to an existing box on my garage. After removing the old light, I notice it only has white/black wiring while the new light fixture has of course white/black/green (ground) wires to attach. What do I do if no ground wire is available with the old box? $M$ finitely generated if submodule and quotient are finitely generated. I know this must be easy. I'm new to modules, so perhaps I must be missing  something important.     Let $M$ be an $R$-module. Show that if there exists a submodule $N$ such that $N$ and $M/N$ are finitely generated, then $M$ is finitely generated.   If the exact sequence $0 \longrightarrow N \longrightarrow M \longrightarrow M/N \longrightarrow 0 $ (where the second and third arrows are inclusion and projection respectively) splits then $M$ would be isomorphic to the direct sum of $M$ and $M/N$ and the conclusion would follow (I think). But I can't show that it splits.  Thank you. How to write a query for exact pairs of groups Prove that $$1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6 \cdot 9}+.........=\sqrt{3}$$   $\bf{My\; Try::}$ Using Binomial expansion of  $$(1-x)^{-n} = 1+nx+\frac{n(n+1)}{2}x^2+\frac{n(n+1)(n+2)}{6}x^3+.......$$  So we get $$nx=\frac{1}{3}$$ and $$\frac{nx(nx+x)}{2}=\frac{1}{3}\cdot \frac{3}{6}$$  We get $$\frac{1}{3}\left(\frac{1}{3}+x\right)=\frac{1}{3}\Rightarrow x=\frac{2}{3}$$  So we get $$n=\frac{1}{2}$$  So our series sum is $$(1-x)^{-n} = \left(1-\frac{2}{3}\right)^{-\frac{1}{2}} = \sqrt{3}$$  Although I know that this is the simplest proof, can we solve it any other way someting like defining $a_{n}$ and then use Telescopic sum.  Thanks. Does "Going Infinite" necessarily end the game? I once heard the late John Ciardi (NPR's "A Word in Your Ear") try to explain that the 1920s idiom, "copacetic" (meaning completely satisfactory), entered into the African-American vocabulary in Harlem from the days when Jews and African Americans lived there together.  He argued that copacetic has the same meaning as the Israeli idiom "kol b'seder" which literally means "all is in order."  The problem with that, said my Harlem-raised father-in-law, is that the Jews in Harlem spoke Yiddish and kol b'seder was not used in Yiddish.  The dictionary I've got is not helpful. Can someone come up with a better explanation? Can't open a folder in windows 7 64bit, by double clicking, it shows me that the default option for opening folder is "cmd" (it's highlighted when i'm right clicking a folder, and the "open" option comes next). I've tried running some registry fixing tools, but it didn't fix it. Can someone suggest me what should I do. I have not had to mess with mailto links much.  However I now need to add a link in the body of a mailto if it is possible.    Is there a way to add a link or to change the email opened to an html email vs a text email?  Something like:  &lt;a href="mailto:test@test.test?body=The message's first paragraph.%0A%0aSecond paragraph.%0A%0AThird Paragraph.%0A%0ALink goes here"&gt;Link text goes here&lt;/a&gt;