Find the density function $X+Y$, where $X$ and $Y$ have a joint density $f(x,y)=lambda^2exp(-lambda y)$.












0












$begingroup$



Find the density function $X+Y$, where $X$ and $Y$ have a joint density $f(x,y)=lambda^2e^{-lambda y}$ for $0leq x leq y, lambda >0 $.




I used the Density transformation theorem to find the joint density of $g(s,t)= lambda^2 e^{-lambda t}$.



Then I find the marginal density $g(s)= int g(s,t) dt$ where $s= x + y$, and $t = y$.



However, I struggle with two things:




  1. How to determine the range of $g(s,t)$ function?

  2. How to determine the lower and upper boundry pf the integral?






probability-theory statistics probability-distributions definite-integrals marginal-distribution







share|cite|improve this question











$endgroup$












  • $begingroup$
    Why do you need the range of the density?
    $endgroup$
    – orange
    Dec 14 '18 at 22:02










  • $begingroup$
    @orange is that not the "right way" to always write a density together with its range? Also, if i am not mistaken, the two question are related. The boundaries of the integral can be determined from the range of the distribution? Of course, I am asking this questions because i dont know any better
    $endgroup$
    – user1607
    Dec 14 '18 at 22:04












  • $begingroup$
    Well, for the range, the function is independent of $x$ and is continuous. So get the limit for $yto 0$ and $yto infty$.
    $endgroup$
    – orange
    Dec 14 '18 at 22:18












  • $begingroup$
    But again, you integrate over the domain and not the range.
    $endgroup$
    – orange
    Dec 14 '18 at 22:25






  • 1




    $begingroup$
    "is that not the "right way" to always write a density together with its range?" Indeed, and this is why the writing in your title, say, is misleading. You should get used to write PDFs correctly, that is, their range included, here, $$f(x,y)=lambda^2e^{-lambda y}mathbf 1_{0<x<y}$$ and the rest of your questions becomes automatically trivial...
    $endgroup$
    – Did
    Dec 15 '18 at 10:15
















0












$begingroup$



Find the density function $X+Y$, where $X$ and $Y$ have a joint density $f(x,y)=lambda^2e^{-lambda y}$ for $0leq x leq y, lambda >0 $.




I used the Density transformation theorem to find the joint density of $g(s,t)= lambda^2 e^{-lambda t}$.



Then I find the marginal density $g(s)= int g(s,t) dt$ where $s= x + y$, and $t = y$.



However, I struggle with two things:




  1. How to determine the range of $g(s,t)$ function?

  2. How to determine the lower and upper boundry pf the integral?






probability-theory statistics probability-distributions definite-integrals marginal-distribution







share|cite|improve this question











$endgroup$












  • $begingroup$
    Why do you need the range of the density?
    $endgroup$
    – orange
    Dec 14 '18 at 22:02










  • $begingroup$
    @orange is that not the "right way" to always write a density together with its range? Also, if i am not mistaken, the two question are related. The boundaries of the integral can be determined from the range of the distribution? Of course, I am asking this questions because i dont know any better
    $endgroup$
    – user1607
    Dec 14 '18 at 22:04












  • $begingroup$
    Well, for the range, the function is independent of $x$ and is continuous. So get the limit for $yto 0$ and $yto infty$.
    $endgroup$
    – orange
    Dec 14 '18 at 22:18












  • $begingroup$
    But again, you integrate over the domain and not the range.
    $endgroup$
    – orange
    Dec 14 '18 at 22:25






  • 1




    $begingroup$
    "is that not the "right way" to always write a density together with its range?" Indeed, and this is why the writing in your title, say, is misleading. You should get used to write PDFs correctly, that is, their range included, here, $$f(x,y)=lambda^2e^{-lambda y}mathbf 1_{0<x<y}$$ and the rest of your questions becomes automatically trivial...
    $endgroup$
    – Did
    Dec 15 '18 at 10:15














0












0








0


0



$begingroup$



Find the density function $X+Y$, where $X$ and $Y$ have a joint density $f(x,y)=lambda^2e^{-lambda y}$ for $0leq x leq y, lambda >0 $.




I used the Density transformation theorem to find the joint density of $g(s,t)= lambda^2 e^{-lambda t}$.



Then I find the marginal density $g(s)= int g(s,t) dt$ where $s= x + y$, and $t = y$.



However, I struggle with two things:




  1. How to determine the range of $g(s,t)$ function?

  2. How to determine the lower and upper boundry pf the integral?






probability-theory statistics probability-distributions definite-integrals marginal-distribution







share|cite|improve this question











$endgroup$





Find the density function $X+Y$, where $X$ and $Y$ have a joint density $f(x,y)=lambda^2e^{-lambda y}$ for $0leq x leq y, lambda >0 $.




I used the Density transformation theorem to find the joint density of $g(s,t)= lambda^2 e^{-lambda t}$.



Then I find the marginal density $g(s)= int g(s,t) dt$ where $s= x + y$, and $t = y$.



However, I struggle with two things:




  1. How to determine the range of $g(s,t)$ function?

  2. How to determine the lower and upper boundry pf the integral?






probability-theory statistics probability-distributions definite-integrals marginal-distribution




probability-theory statistics probability-distributions definite-integrals marginal-distribution






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 15 '18 at 10:04









Davide Giraudo

127k16152267




127k16152267










asked Dec 14 '18 at 21:57









user1607user1607

1718




1718












  • $begingroup$
    Why do you need the range of the density?
    $endgroup$
    – orange
    Dec 14 '18 at 22:02










  • $begingroup$
    @orange is that not the "right way" to always write a density together with its range? Also, if i am not mistaken, the two question are related. The boundaries of the integral can be determined from the range of the distribution? Of course, I am asking this questions because i dont know any better
    $endgroup$
    – user1607
    Dec 14 '18 at 22:04












  • $begingroup$
    Well, for the range, the function is independent of $x$ and is continuous. So get the limit for $yto 0$ and $yto infty$.
    $endgroup$
    – orange
    Dec 14 '18 at 22:18












  • $begingroup$
    But again, you integrate over the domain and not the range.
    $endgroup$
    – orange
    Dec 14 '18 at 22:25






  • 1




    $begingroup$
    "is that not the "right way" to always write a density together with its range?" Indeed, and this is why the writing in your title, say, is misleading. You should get used to write PDFs correctly, that is, their range included, here, $$f(x,y)=lambda^2e^{-lambda y}mathbf 1_{0<x<y}$$ and the rest of your questions becomes automatically trivial...
    $endgroup$
    – Did
    Dec 15 '18 at 10:15


















  • $begingroup$
    Why do you need the range of the density?
    $endgroup$
    – orange
    Dec 14 '18 at 22:02










  • $begingroup$
    @orange is that not the "right way" to always write a density together with its range? Also, if i am not mistaken, the two question are related. The boundaries of the integral can be determined from the range of the distribution? Of course, I am asking this questions because i dont know any better
    $endgroup$
    – user1607
    Dec 14 '18 at 22:04












  • $begingroup$
    Well, for the range, the function is independent of $x$ and is continuous. So get the limit for $yto 0$ and $yto infty$.
    $endgroup$
    – orange
    Dec 14 '18 at 22:18












  • $begingroup$
    But again, you integrate over the domain and not the range.
    $endgroup$
    – orange
    Dec 14 '18 at 22:25






  • 1




    $begingroup$
    "is that not the "right way" to always write a density together with its range?" Indeed, and this is why the writing in your title, say, is misleading. You should get used to write PDFs correctly, that is, their range included, here, $$f(x,y)=lambda^2e^{-lambda y}mathbf 1_{0<x<y}$$ and the rest of your questions becomes automatically trivial...
    $endgroup$
    – Did
    Dec 15 '18 at 10:15
















$begingroup$
Why do you need the range of the density?
$endgroup$
– orange
Dec 14 '18 at 22:02




$begingroup$
Why do you need the range of the density?
$endgroup$
– orange
Dec 14 '18 at 22:02












$begingroup$
@orange is that not the "right way" to always write a density together with its range? Also, if i am not mistaken, the two question are related. The boundaries of the integral can be determined from the range of the distribution? Of course, I am asking this questions because i dont know any better
$endgroup$
– user1607
Dec 14 '18 at 22:04






$begingroup$
@orange is that not the "right way" to always write a density together with its range? Also, if i am not mistaken, the two question are related. The boundaries of the integral can be determined from the range of the distribution? Of course, I am asking this questions because i dont know any better
$endgroup$
– user1607
Dec 14 '18 at 22:04














$begingroup$
Well, for the range, the function is independent of $x$ and is continuous. So get the limit for $yto 0$ and $yto infty$.
$endgroup$
– orange
Dec 14 '18 at 22:18






$begingroup$
Well, for the range, the function is independent of $x$ and is continuous. So get the limit for $yto 0$ and $yto infty$.
$endgroup$
– orange
Dec 14 '18 at 22:18














$begingroup$
But again, you integrate over the domain and not the range.
$endgroup$
– orange
Dec 14 '18 at 22:25




$begingroup$
But again, you integrate over the domain and not the range.
$endgroup$
– orange
Dec 14 '18 at 22:25




1




1




$begingroup$
"is that not the "right way" to always write a density together with its range?" Indeed, and this is why the writing in your title, say, is misleading. You should get used to write PDFs correctly, that is, their range included, here, $$f(x,y)=lambda^2e^{-lambda y}mathbf 1_{0<x<y}$$ and the rest of your questions becomes automatically trivial...
$endgroup$
– Did
Dec 15 '18 at 10:15




$begingroup$
"is that not the "right way" to always write a density together with its range?" Indeed, and this is why the writing in your title, say, is misleading. You should get used to write PDFs correctly, that is, their range included, here, $$f(x,y)=lambda^2e^{-lambda y}mathbf 1_{0<x<y}$$ and the rest of your questions becomes automatically trivial...
$endgroup$
– Did
Dec 15 '18 at 10:15










1 Answer
1






active

oldest

votes


















0












$begingroup$

The pdf of $Z:=X+Y$ is $$int_{0le xle y}lambda^2 e^{-lambda y}delta (x+y-z)dxdy=int_{z/2}^zlambda^2 e^{-lambda y}dy=lambda (e^{-lambda z/2}-e^{-lambda z})$$for $z=0$. (Note this is clearly a pdf.) I'll explain the steps below.



First we write down an integral using the $delta$ function; then we integrate out $x$, which only retains value of $y$ for which $x+y-z$ is negative when $x=0$ but positive when $x=y$, so that the integral is around the delta peak. These constraints can be stated as $yle zle 2y$, or equivalently $y/2le zle y$.






share|cite|improve this answer









$endgroup$













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039925%2ffind-the-density-function-xy-where-x-and-y-have-a-joint-density-fx-y%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0












    $begingroup$

    The pdf of $Z:=X+Y$ is $$int_{0le xle y}lambda^2 e^{-lambda y}delta (x+y-z)dxdy=int_{z/2}^zlambda^2 e^{-lambda y}dy=lambda (e^{-lambda z/2}-e^{-lambda z})$$for $z=0$. (Note this is clearly a pdf.) I'll explain the steps below.



    First we write down an integral using the $delta$ function; then we integrate out $x$, which only retains value of $y$ for which $x+y-z$ is negative when $x=0$ but positive when $x=y$, so that the integral is around the delta peak. These constraints can be stated as $yle zle 2y$, or equivalently $y/2le zle y$.






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      The pdf of $Z:=X+Y$ is $$int_{0le xle y}lambda^2 e^{-lambda y}delta (x+y-z)dxdy=int_{z/2}^zlambda^2 e^{-lambda y}dy=lambda (e^{-lambda z/2}-e^{-lambda z})$$for $z=0$. (Note this is clearly a pdf.) I'll explain the steps below.



      First we write down an integral using the $delta$ function; then we integrate out $x$, which only retains value of $y$ for which $x+y-z$ is negative when $x=0$ but positive when $x=y$, so that the integral is around the delta peak. These constraints can be stated as $yle zle 2y$, or equivalently $y/2le zle y$.






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        The pdf of $Z:=X+Y$ is $$int_{0le xle y}lambda^2 e^{-lambda y}delta (x+y-z)dxdy=int_{z/2}^zlambda^2 e^{-lambda y}dy=lambda (e^{-lambda z/2}-e^{-lambda z})$$for $z=0$. (Note this is clearly a pdf.) I'll explain the steps below.



        First we write down an integral using the $delta$ function; then we integrate out $x$, which only retains value of $y$ for which $x+y-z$ is negative when $x=0$ but positive when $x=y$, so that the integral is around the delta peak. These constraints can be stated as $yle zle 2y$, or equivalently $y/2le zle y$.






        share|cite|improve this answer









        $endgroup$



        The pdf of $Z:=X+Y$ is $$int_{0le xle y}lambda^2 e^{-lambda y}delta (x+y-z)dxdy=int_{z/2}^zlambda^2 e^{-lambda y}dy=lambda (e^{-lambda z/2}-e^{-lambda z})$$for $z=0$. (Note this is clearly a pdf.) I'll explain the steps below.



        First we write down an integral using the $delta$ function; then we integrate out $x$, which only retains value of $y$ for which $x+y-z$ is negative when $x=0$ but positive when $x=y$, so that the integral is around the delta peak. These constraints can be stated as $yle zle 2y$, or equivalently $y/2le zle y$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 15 '18 at 10:15









        J.G.J.G.

        28.7k22845




        28.7k22845






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039925%2ffind-the-density-function-xy-where-x-and-y-have-a-joint-density-fx-y%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Bundesstraße 106

            Image
            Dieser Artikel beschreibt die Bundesstraße 106 in Deutschland. Zur gleichnamigen Straße in Österreich siehe Mölltal Straße. Vorlage:Infobox hochrangige Straße/Wartung/DE-B Bundesstraße 106 in Deutschland Karte Basisdaten Betreiber: Deutschland  Bundesrepublik Deutschland Straßenbeginn: Wismar ( 53° 54′  N , 11° 24′  O 53.900536 11.396875 ) Straßenende: Schwerin ( 53° 36′  N , 11° 24′  O 53.592785 11.404892 ) Gesamtlänge: 40,7 km Bundesland: Mecklenburg-Vorpommern Ausbauzustand: zweistreifig [1] Bundesstraße 106 in Schwerin (Umgehungsstraße) Straßenverlauf Land Mecklenburg-Vorpommern Landkreis Nordwestmecklenburg Ortsumgehung Wismar Wallensteingraben über   Dorf Mecklenburg Wallensteingraben Groß Stieten Bad Kleinen OT Niendorf Bahnstrecke Lübeck–Bad Kleinen Zickhusen Lübstorf Klein Trebbow OT Kirch Stück ...
            Read more

            Verónica Boquete

            Image
            Verónica Boquete im Trikot der Nationalmannschaft (2013) Personalia Name Verónica Boquete Giadans Geburtstag 9. April 1987 Geburtsort Santiago de Compostela, Spanien Größe 159 cm Position Mittelfeld / Sturm Juniorinnen Jahre Station 2001–2005 Xuventú Aguiño Frauen Jahre Station Spiele (Tore) 1 2005–2008 Prainsa Zaragoza 83 (40) 2008–2010 Espanyol Barcelona 53 (30) 2010 Buffalo Flash 9 0 (9) 2010 Chicago Red Stars 3 0 (1) 2010–2011 Espanyol Barcelona 26 (41) 2011 Philadelphia Independence 13 0 (5) 2011 Energija Woronesch 5 0 (1) 2012–2014 Tyresö FF 39 (14) 2014 Portland Thorns FC 15 0 (4) 2014–2015 1. FFC Frankfurt 23 0 (7) 2015–2016 FC Bayern München 9 0 (0) 2016–2018 Paris Saint-Germain 31 0 (8) 2018 Peking BG Phoenix 2019– Utah Royals FC 0 0 (0) Nationalmannschaft Jahre Auswahl ...
            Read more

            Ida-Boy-Ed-Garten

            Image
            Die Lage des Ida-Boy-Ed-Gartens, rot markiert Der Ida-Boy-Ed-Garten Der Ida-Boy-Ed-Garten, Blick von der Wakenitzmauer durch das Tor in der Stadtmauer Der Ida-Boy-Ed-Garten ist eine Straße der Lübecker Altstadt. Inhaltsverzeichnis 1 Lage 2 Geschichte 3 Literatur 4 Weblinks Lage | Der etwa 90 Meter lange Ida-Boy-Ed-Garten befindet sich an der nördlichen Spitze der Altstadtinsel, vor dem Burgtor. Die in einem Bogen durch eine Grünanlage hangaufwärts führende Straße verbindet die Wakenitzmauer mit der Burgtorbrücke. Geschichte | Der heutige Ida-Boy-Ed-Garten wurde ursprünglich als Verlängerung der Wakenitzmauer angelegt. 1903 wurde in den erhaltenen Teil der mittelalterlichen Stadtmauer, an dem die Wakenitzmauer bei der Einmündung der Kaiserstraße begann, ein Tor mit einem großen Bogen für die Durchführung der Straße und zwei kleineren seitlichen Öffnungen für Fußgänger gebrochen, um für die neu errichteten Wohnhäuser in diesem Bereich ...
            Read more