$mathbb{E}[X] ,Var[X] $ of quadratic equation












1












$begingroup$


I am having some problems to find the solution required in the next exercise:




$x^2-ax+b=0$ has real and not equal solutions and that b is a positive constant that such that $b sim U(i,j)$ for certain i,j.



Find the $mathbb{E}[X], ,Var[X]$ of the solutions.





What I have so far:



From the Quadratic Formula, it is known that:



And since these are real solutions,



$implies a^2>4b$



$implies frac{a^2}{4}>b$




¿how do I use $b sim U(i,j)$?






probability probability-theory statistics probability-distributions normal-distribution







share|cite|improve this question











$endgroup$












  • $begingroup$
    Please do not deface the question after it's answered.
    $endgroup$
    – Saad
    Dec 6 '18 at 3:08
















1












$begingroup$


I am having some problems to find the solution required in the next exercise:




$x^2-ax+b=0$ has real and not equal solutions and that b is a positive constant that such that $b sim U(i,j)$ for certain i,j.



Find the $mathbb{E}[X], ,Var[X]$ of the solutions.





What I have so far:



From the Quadratic Formula, it is known that:



And since these are real solutions,



$implies a^2>4b$



$implies frac{a^2}{4}>b$




¿how do I use $b sim U(i,j)$?






probability probability-theory statistics probability-distributions normal-distribution







share|cite|improve this question











$endgroup$












  • $begingroup$
    Please do not deface the question after it's answered.
    $endgroup$
    – Saad
    Dec 6 '18 at 3:08














1












1








1


1



$begingroup$


I am having some problems to find the solution required in the next exercise:




$x^2-ax+b=0$ has real and not equal solutions and that b is a positive constant that such that $b sim U(i,j)$ for certain i,j.



Find the $mathbb{E}[X], ,Var[X]$ of the solutions.





What I have so far:



From the Quadratic Formula, it is known that:



And since these are real solutions,



$implies a^2>4b$



$implies frac{a^2}{4}>b$




¿how do I use $b sim U(i,j)$?






probability probability-theory statistics probability-distributions normal-distribution







share|cite|improve this question











$endgroup$




I am having some problems to find the solution required in the next exercise:




$x^2-ax+b=0$ has real and not equal solutions and that b is a positive constant that such that $b sim U(i,j)$ for certain i,j.



Find the $mathbb{E}[X], ,Var[X]$ of the solutions.





What I have so far:



From the Quadratic Formula, it is known that:



And since these are real solutions,



$implies a^2>4b$



$implies frac{a^2}{4}>b$




¿how do I use $b sim U(i,j)$?






probability probability-theory statistics probability-distributions normal-distribution




probability probability-theory statistics probability-distributions normal-distribution






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 6 '18 at 3:27







Israel Barquín

















asked Dec 5 '18 at 23:03









Israel BarquínIsrael Barquín

276




276












  • $begingroup$
    Please do not deface the question after it's answered.
    $endgroup$
    – Saad
    Dec 6 '18 at 3:08


















  • $begingroup$
    Please do not deface the question after it's answered.
    $endgroup$
    – Saad
    Dec 6 '18 at 3:08
















$begingroup$
Please do not deface the question after it's answered.
$endgroup$
– Saad
Dec 6 '18 at 3:08




$begingroup$
Please do not deface the question after it's answered.
$endgroup$
– Saad
Dec 6 '18 at 3:08










1 Answer
1






active

oldest

votes


















1












$begingroup$

Well, simply by using the fact that since $b sim U(0,a^2/4)$ then $mathbb{E}(b) =4/a^2int_{0}^{a^2/2}b,db =frac{a^2/4}{2} = a^2/8$.



Then, for $x_1$ for example, it is :



$$mathbb{E}(x_1) = mathbb{E}left(frac{a + sqrt{a^2-4b}}{2}right) = frac{1}{2}mathbb{E}left(a+sqrt{a^2-4b}right)$$



But, $a$ is a constant (not following any distributions, as far as one can say per the problem hypothesis), thus $mathbb{E}(a) = a$ and so :



$$mathbb{E}(x_1) = frac{1}{2}a + frac{1}{2}mathbb{E}left(sqrt{a^2-4b}right) $$



Can you finish the calculation now ?



For the calculation of variance, again use the fact that since $b sim U(0,a^2/4)$ then :



$$V(b) = frac{1}{12}left(frac{a^2}{4}-0right)^2= frac{a^4}{12cdot 16}$$



Be careful now, since if $a$ is a constant, then $V(a) = a^2$.






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%2f3027786%2fmathbbex-varx-of-quadratic-equation%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









    1












    $begingroup$

    Well, simply by using the fact that since $b sim U(0,a^2/4)$ then $mathbb{E}(b) =4/a^2int_{0}^{a^2/2}b,db =frac{a^2/4}{2} = a^2/8$.



    Then, for $x_1$ for example, it is :



    $$mathbb{E}(x_1) = mathbb{E}left(frac{a + sqrt{a^2-4b}}{2}right) = frac{1}{2}mathbb{E}left(a+sqrt{a^2-4b}right)$$



    But, $a$ is a constant (not following any distributions, as far as one can say per the problem hypothesis), thus $mathbb{E}(a) = a$ and so :



    $$mathbb{E}(x_1) = frac{1}{2}a + frac{1}{2}mathbb{E}left(sqrt{a^2-4b}right) $$



    Can you finish the calculation now ?



    For the calculation of variance, again use the fact that since $b sim U(0,a^2/4)$ then :



    $$V(b) = frac{1}{12}left(frac{a^2}{4}-0right)^2= frac{a^4}{12cdot 16}$$



    Be careful now, since if $a$ is a constant, then $V(a) = a^2$.






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      Well, simply by using the fact that since $b sim U(0,a^2/4)$ then $mathbb{E}(b) =4/a^2int_{0}^{a^2/2}b,db =frac{a^2/4}{2} = a^2/8$.



      Then, for $x_1$ for example, it is :



      $$mathbb{E}(x_1) = mathbb{E}left(frac{a + sqrt{a^2-4b}}{2}right) = frac{1}{2}mathbb{E}left(a+sqrt{a^2-4b}right)$$



      But, $a$ is a constant (not following any distributions, as far as one can say per the problem hypothesis), thus $mathbb{E}(a) = a$ and so :



      $$mathbb{E}(x_1) = frac{1}{2}a + frac{1}{2}mathbb{E}left(sqrt{a^2-4b}right) $$



      Can you finish the calculation now ?



      For the calculation of variance, again use the fact that since $b sim U(0,a^2/4)$ then :



      $$V(b) = frac{1}{12}left(frac{a^2}{4}-0right)^2= frac{a^4}{12cdot 16}$$



      Be careful now, since if $a$ is a constant, then $V(a) = a^2$.






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        Well, simply by using the fact that since $b sim U(0,a^2/4)$ then $mathbb{E}(b) =4/a^2int_{0}^{a^2/2}b,db =frac{a^2/4}{2} = a^2/8$.



        Then, for $x_1$ for example, it is :



        $$mathbb{E}(x_1) = mathbb{E}left(frac{a + sqrt{a^2-4b}}{2}right) = frac{1}{2}mathbb{E}left(a+sqrt{a^2-4b}right)$$



        But, $a$ is a constant (not following any distributions, as far as one can say per the problem hypothesis), thus $mathbb{E}(a) = a$ and so :



        $$mathbb{E}(x_1) = frac{1}{2}a + frac{1}{2}mathbb{E}left(sqrt{a^2-4b}right) $$



        Can you finish the calculation now ?



        For the calculation of variance, again use the fact that since $b sim U(0,a^2/4)$ then :



        $$V(b) = frac{1}{12}left(frac{a^2}{4}-0right)^2= frac{a^4}{12cdot 16}$$



        Be careful now, since if $a$ is a constant, then $V(a) = a^2$.






        share|cite|improve this answer











        $endgroup$



        Well, simply by using the fact that since $b sim U(0,a^2/4)$ then $mathbb{E}(b) =4/a^2int_{0}^{a^2/2}b,db =frac{a^2/4}{2} = a^2/8$.



        Then, for $x_1$ for example, it is :



        $$mathbb{E}(x_1) = mathbb{E}left(frac{a + sqrt{a^2-4b}}{2}right) = frac{1}{2}mathbb{E}left(a+sqrt{a^2-4b}right)$$



        But, $a$ is a constant (not following any distributions, as far as one can say per the problem hypothesis), thus $mathbb{E}(a) = a$ and so :



        $$mathbb{E}(x_1) = frac{1}{2}a + frac{1}{2}mathbb{E}left(sqrt{a^2-4b}right) $$



        Can you finish the calculation now ?



        For the calculation of variance, again use the fact that since $b sim U(0,a^2/4)$ then :



        $$V(b) = frac{1}{12}left(frac{a^2}{4}-0right)^2= frac{a^4}{12cdot 16}$$



        Be careful now, since if $a$ is a constant, then $V(a) = a^2$.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 6 '18 at 2:21









        Israel Barquín

        276




        276










        answered Dec 5 '18 at 23:16









        RebellosRebellos

        14.5k31246




        14.5k31246






























            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%2f3027786%2fmathbbex-varx-of-quadratic-equation%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

            To store a contact into the json file from server.js file using a class in NodeJS

            Image
            0 I have created three files -contactPresentation.js, contactService.js, contacts.json. I will run the contactService.js file. And it will navigate to contactPresentation.js file that will only choose an option to either add or display a file. If add(), then add the data into add of service file.And the data will be added to the contacts.json file . And if display(), then display the display() of service file. And the display will take the data from contacts.json file. However, i know my code is flawed. Please correct it . I am also getting an error - this.contacts.push(new ContactService(userName, contactNumber, emailId)); ^ TypeError: Cannot read property 'push' of undefined at ContactService.add (e:NodeJSProjectcontactManager2contactService.js:57:23) at new C
            Read more

            Redirect URL with Chrome Remote Debugging Android Devices

            Image
            1 How can i redirect an URL with Chrome Remote Debugging Android Devices? like Requestly Chrome extension on desktop. google-chrome debugging url redirect testing share | improve this question edited Nov 24 '18 at 8:16 sachinjain024 14.7k 23 73 129 asked Nov 23 '18 at 16:09 Momen Momen 91 10
            Read more

            Dieringhausen

            Image
            Dieringhausen Stadt Gummersbach 50.982777777778 7.5316666666667 165 Koordinaten: 50° 58′ 58″  N , 7° 31′ 54″  O Höhe: 165  (160–245)  m Einwohner: 5055  (30. Jun. 2016) Postleitzahl: 51645 Vorwahl: 02261 Lage von Dieringhausen in Gummersbach Im Hohl ca. 1927–1930 Ansicht von Südwesten bei Bünghausen Dieringhausen (im örtlichen Dialekt Dierkesen ) ist ein Ortsteil von Gummersbach im Oberbergischen Kreis, Regierungsbezirk Köln, Nordrhein-Westfalen (Deutschland). Inhaltsverzeichnis 1 Geographie 2 Geschichte 3 Kultur 3.1 Sehenswürdigkeiten 3.2 Regelmäßige Veranstaltungen 4 Infrastruktur und Wirtschaft 4.1 Verkehr 4.1.1 Schienen- und Busverkehr 4.1.2 Straßen 4.2 Öffentliche Einrichtungen 4.2.1 Schulen und Bildungseinrichtungen 4.2.2 Kirchliche Einrichtungen 5 Persönlichkeiten 5.1 In Dieringhausen geboren 5.2 In Dieringhausen gelebt 6 Einzelnachweise 7
            Read more