How can I find the marginal distributions of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$? Let $X_1 \sim Gamma(\alpha_1,1)$ and $X_2 \sim Gamma(\alpha_2,1)$ be independent random variables. How can I find the marginal distributions of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$?  By setting $U=\frac{X_1}{X_1+X_2}$ and $V=\frac{X_2}{X_1+X_2}$, I am having a hard time trying to isolate $X_1$ and $X_2$. Once I can isolate these variables, I can complete the problem. However, I was wondering if there was a better way to solve for those marginal distributions.
Construction of Dirichlet distribution with Gamma distribution Let $X_1,\dots,X_{k+1}$ be mutually independent random variables, each having a gamma distribution with parameters $\alpha_i,i=1,2,\dots,k+1$ show that $Y_i=\frac{X_i}{X_1+\cdots+X_{k+1}},i=1,\dots,k$, have a joint ditribution as $\text{Dirichlet}(\alpha_1,\alpha_2,\dots,\alpha_k;\alpha_{k+1})$   Joint pdf of $(X_1,\dots,X_{k+1})=\frac{e^{-\sum_{i=1}^{k+1}x_i}x_1^{\alpha_1-1}\dots x_{k+1}^{\alpha_{k+1}-1}}{\Gamma(\alpha_1)\Gamma(\alpha_2)\dots \Gamma(\alpha_{k+1})}$.Then to find joint pdf of $(Y_1,\dots,Y_{k+1})$ I can not find jacobian i.e.$J(\frac{x_1,\dots,x_{k+1}}{y_1,\dots,y_{k+1}})$
What's the best way to find the inverse of datetime.isocalendar()? This is an idea I have seen around, but I have not seen a feature request for it, so here it is.  Downvotes that have no comments are frustrating and difficult to manage.  This is a way to help improve that:   Have a downvote initially cost more to cast for the downvoter (5 points for example) When a comment (from the downvoter) on the downvoted question/answer gets an upvote, you get your points back.   All points (ie all 5) could be refunded or just up till the traditional -1 that is in place now (ie 4 points refunded).    This would encourage good comments and downvotes for real reasons.  (Because comments for questions/answers that are bias related will not get the upvotes as readily.)  This could/should be limited to non-subjective tags.  (ie tags where a downvote just because you "don't like it" is not appropriate.) My cat sat on my laptop and now if I type either L or P, it inputs 3 or * instead. The rest of my keyboard still works correctly. The problem is bad enough that I had to use the touch screen of my computer to type this in.  How can I restore the correct working behaviour of my keyboard? Am I being scammed by a sugar daddy? General theorems for consistency and asymptotic normality of maximum likelihood I'm interested in a good reference for results concerning asymptotic properties of maximum likelihood estimators. Consider a model $\{f_n(\cdot \mid \theta): \theta \in \Theta, n \in \mathbb N\}$ where $f_n(\mathbf x \mid \theta)$ is an $n$-dimensional density, and $\hat \theta_n$ is the MLE based on a sample $X_1, \ldots, X_n$ from $f_n(\cdot \mid \theta_0)$ where $\theta_0$ is the "true" value of $\theta$. There are two irregularities I'm interested in.    The data $X_1, \ldots, X_n$ are not iid and, as a result, the Fisher information about $\theta$ accrues at a rate slower than $n$.  $\Theta$ is a bounded set, and with positive probability $\hat \theta_n$ lies on the boundary. The boundary corresponds to a "simpler" model, and so there is particular interest in whether or not $\theta_0$ lies on the boundary.    My particular questions are   Letting $J_n(\theta)$ denote the observed Fisher information corresponding to $\theta$, and suppose $\theta_0$ lies in the interior of $\Theta$. Under what conditions is $$\left[J_n(\hat \theta_n)\right]^{1/2}(\hat \theta_n - \theta_0)$$ asymptotically normal as $n \to \infty$? In particular, are the regularity conditions similar to the usual ones, with the relevant modification being $J_n(\hat \theta_n) \to \infty$ in some sense? Suppose instead that $\theta_0$ is on the boundary, and again recall that $\hat \theta_n = \theta_0$ happens with positive probability - for concreteness, in a mixed effects model $Y_{ij} = \mu + \beta_i + \epsilon_{ij}$ we can have $\hat \sigma_{\beta}^2 = 0$. Under what conditions does $\hat \theta_n \to \theta_0$ (almost surely or in probability) and under what conditions will $\hat \theta_n = \theta_0$ eventually (this probably fails for the mixed effects model, but corresponds to "oracle" properties for the LASSO and related estimators, so perhaps it is too much to ask for general results)?   Again, just a pointer to a text with results at this level of generality would be greatly appreciated. Python module for converting PDF to text Is there any python module to convert PDF files into text? I tried  found in Activestate which uses pypdf but the text generated had no space between and was of no use. A zero-dimensional ring is Noetherian? Proposition: Let $A$ be a non-zero ring that is not a field. Suppose $A$ is zero dimensional. Then it is Noetherian.   Proof: Let $p$ be a prime ideal of $A$. If $p$ is not maximal, then $p \subsetneq m$ for a maximal ideal $m$. Hence $dim A \ge 1$, contradiction. Hence every prime ideal is maximal. Consequently, every increasing (or decreasing) chain of prime ideals will be trivial in the sense that it will have length $0$. Hence the ring is Noetherian (and Artinian).  Is there a problem with the above proof? How to format numbers as currency strings Can't convert string into float/double What are these stamps in my passport after I was advised to divert via the UK but didn't have a visa? I am from Myanmar. I was on my way to Venice from Brussels while holding Schengen visa (single entry). I bought a British Airways ticket to fly from Brussels to Venice. Unfortunately, on that day, all BA flights from Brussels were canceled. British Airway's customer service centre told me to go to from Brussels to London by train, where I could catch my flight to Venice. I took a taxi to Bruxelles-Midi train station and bought a ticket to London. A Brussels immigration officer inspected and approved my passport.   Then I had to to through UK Border Immigration. There, they said I can't enter to get my flight. Although I explained the situation with the airline and made clear of my destination (I already reserved my hotels in Venice, Switzerland and France as described in my travel itinerary), they detained me for some hours and also took my fingerprints on paper and photo.   They asked questions concerned about my finances and gave me a refusal letter for not having a transit visa. So I had to re-enter Brussels where the immigration officer put the below stamp in my passport. As I had a single entry Schengen visa, I had to buy another airline ticket to Venice and complete my Europe trip.   What do the stamps below mean on my passport?   Can they affect me and, if so, how and why, when I apply for another Schengen visa? GET and POST methods with the same Action name in the same Controller How to find the shortest path between opposite vertices of a cube, traveling on its surface? I am stuck with the following problem that says:      Let $A,B$ be the ends of the longest diagonal of the unit cube . The length of    the shortest path from $A$ to $B$ along the surface is :     $\sqrt{3}\,\,$  2.$\,\,1+\sqrt{2}\,\,$  3.$\,\,\sqrt{5}\,\,$ 4.$\,\,3$    My Try:   So, the length of the longest diagonal $AB=\sqrt{3}$. If I reach from $A$ to $B$ along the surface line $AC+CD+BD$, then it gives $3$ units. But the answer is given to be option 3.   Can someone explain? Thanks in advance for your time. How to add a script to Startup Applications from the command line? I have a shell script. I want to configure my script to run automatically during startup. I know how to do this from GUI. But I wanted to do this from terminal. How can I do this?  On a primary research I found that the file needs to be moved to /etc/int.d/ directory. But this operation needs sudo permission. I wanted to do this without super user permissions.   I also found that there are files under ~/.config/autostart/ which are having some settings regarding the startup applications. But I don't know how to edit them to achieve this.   Can someone tell me the exact procedure to achieve this? How to print fields value in views-view.html.twig in drupal 8 How to print fields value in views-view.html.twig I have created a views-view.html.twig file for my view slider_3. The template name is 'views-view--slider_3.html.twig' The view formatting is done by this template, but I need a field is come at begging position of the view:  &lt;div class="slider-3-main"&gt;{{ rows.required_field }} &lt;div{{ attributes.addClass(classes) }}&gt;   {{ title_prefix }}   {% if title %}     {{ title }}   {% endif %}   {{ title_suffix }}    How can be the field will written to display at {{ rows.required_field }} I've been modeling a hermit crab and I don't know why but some areas of the mesh are really dark on some spots, like the legs and the claws.   I hope you can help me fix it. I'm very new to Ubuntu, and have come to this point that I can't boot back to my Windows Vista anymore.  I thought that I would get some kind of dual boot in the beginning when booting my computer, but this doesn't seem too happen and I don't have the slightest idea on how to fix this. I was looking for help here at askubuntu, but I couldn't seem to find anything. I'm too new with this software.  I'll need hints and tips on how to fix this  Thanks for your time folks! Suppose that $5\leq q\leq p$ are both prime. Prove that $24|(p^2-q^2)$.   This is what I got so far. I figured that since $p,q$ are bigger than $5$, there are only odd primes for this conjecture. Then I can rename $p=2m+1,q=2n+1,m,n∈Z$.  We want to show $(p^2-q^2 )=24k,k∈Z$. By substituting $2m+1$ for $p$ and $2n+1$ for $q$, we get  $((2m+1)^2-(2n+1)^2)$. If we expand it, we get $4(m^2+n^2+m-n)$ Q. I think I can make $4(m^2+n^2+m-n)$ look like $24k$. How should I approach ?  And my prof. commented that "I would take this proof in small steps.  Rather than trying initially to prove that $24 | p^2-q^2$, look at the progress you already have.  You proved $4 | p^2-q^2$.  Can you prove 8 is a factor?  What about 3?  Then see if you can put together what you have to get 24 as a factor."  I am stuck here. Any helps? How can I determine the list of files in a directory from inside my C or C++ code?  I'm not allowed to execute the ls command and parse the results from within my program. LINQ on a LinkedList - iterate over LinkedListNode, not T Replacing non-English characters in attribute tables using ArcPy and Python? I have a few shapefiles where some of the attributes contain the non-English characters ÅÄÖ. Since some queries doesn't work with these characters (specifically ), I tried to change them in advance with a simple script and add the new strings to another field.  However, change in characters works fine but not updating the field with arcpy.UpdateCursor.   What is an appropriate way of solving this?  I have also tried to do this via the Field Calculator while posting "code" to the codeblock, with the same error.  Error message: Runtime error  Traceback (most recent call last):   File "", line 1, in    File "c:/gis/python/teststring.py", line 28, in      val = code(str(prow.Typkod)) UnicodeEncodeError: 'ascii' codec can't encode character u'\xc4' in position 3: ordinal not in range(128)  Code:  # -*- coding: cp1252 -*- def code(infield):     data = ''     for i in infield: ##        print i         if i == 'Ä':             data = data + 'AE'         elif i == 'ä':             data = data + 'ae'         elif i == 'Å':             data = data + 'AA'         elif i == 'å':             data = data + 'aa'         elif i == 'Ö':             data = data + 'OE'         elif i == 'ö':             data = data + 'oe'         else:             data = data + i     return data   shp = r'O:\XXX\250000\DB\ArcView\shape.shp'  prows = arcpy.UpdateCursor(shp)  for prow in prows:     val = code(unicode(str(prow.Typkod), "utf-8"))     prow.Typkod_U = val     print val     prows.updateRow(prow)   The values of Typkod are of the type: [D, D, S, DDRÄ, TRÄ] etc.  I use ArcMap Basic (10.1) on Windows 7.    New Error message:  Runtime error  Traceback (most recent call last):   File "", line 1, in    File "c:/gis/python/teststring.py", line 29, in      val = code(unicode(str(row.Typkod), "utf-8")) UnicodeEncodeError: 'ascii' codec can't encode character u'\xc4' in position 3: ordinal not in range(128)  &gt;&gt;&gt; val 'DDRÄ' &gt;&gt;&gt; type(val) type 'str'      It appears as if the output from the function is wrong somehow. When there's ÅÄÖ involved it returns data = u'DDR\xc4' and not (as was my intention) data = 'DDRAE'. Any suggestions on what might cause this?