Expected number of updates of Maximum

Multi tool use
Multi tool use











up vote
1
down vote

favorite
1












Let $L$ be a list of unique elements. Consider the following standard algorithm for finding the maximum value in $L$:




  1. Initialize the current maximum of the list to be $m = −infty$.

  2. For $i= 1$ up through $n$,check to see if $L[i]>m$; if so, reset $m$ to be $L[i]$.

  3. Output $m$.


Suppose we randomly permuate the elements of $L$ before running the procedure. Calculated the expected number of times $m$ will be reset in Step 2.






probability-theory random-variables foundations expected-value







share|cite|improve this question
























  • Perhaps this, or something like it: cs.stackexchange.com/questions/63682/…
    – Ethan Bolker
    10 hours ago










  • Thank you for the comment. However, my question is about the expected value that the algorithm will output m in list L.
    – Naseeb Thapaliya
    10 hours ago















up vote
1
down vote

favorite
1












Let $L$ be a list of unique elements. Consider the following standard algorithm for finding the maximum value in $L$:




  1. Initialize the current maximum of the list to be $m = −infty$.

  2. For $i= 1$ up through $n$,check to see if $L[i]>m$; if so, reset $m$ to be $L[i]$.

  3. Output $m$.


Suppose we randomly permuate the elements of $L$ before running the procedure. Calculated the expected number of times $m$ will be reset in Step 2.






probability-theory random-variables foundations expected-value







share|cite|improve this question
























  • Perhaps this, or something like it: cs.stackexchange.com/questions/63682/…
    – Ethan Bolker
    10 hours ago










  • Thank you for the comment. However, my question is about the expected value that the algorithm will output m in list L.
    – Naseeb Thapaliya
    10 hours ago













up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1





Let $L$ be a list of unique elements. Consider the following standard algorithm for finding the maximum value in $L$:




  1. Initialize the current maximum of the list to be $m = −infty$.

  2. For $i= 1$ up through $n$,check to see if $L[i]>m$; if so, reset $m$ to be $L[i]$.

  3. Output $m$.


Suppose we randomly permuate the elements of $L$ before running the procedure. Calculated the expected number of times $m$ will be reset in Step 2.






probability-theory random-variables foundations expected-value







share|cite|improve this question















Let $L$ be a list of unique elements. Consider the following standard algorithm for finding the maximum value in $L$:




  1. Initialize the current maximum of the list to be $m = −infty$.

  2. For $i= 1$ up through $n$,check to see if $L[i]>m$; if so, reset $m$ to be $L[i]$.

  3. Output $m$.


Suppose we randomly permuate the elements of $L$ before running the procedure. Calculated the expected number of times $m$ will be reset in Step 2.






probability-theory random-variables foundations expected-value




probability-theory random-variables foundations expected-value






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 hours ago









d.k.o.

8,079527




8,079527










asked 13 hours ago









Naseeb Thapaliya

62




62












  • Perhaps this, or something like it: cs.stackexchange.com/questions/63682/…
    – Ethan Bolker
    10 hours ago










  • Thank you for the comment. However, my question is about the expected value that the algorithm will output m in list L.
    – Naseeb Thapaliya
    10 hours ago


















  • Perhaps this, or something like it: cs.stackexchange.com/questions/63682/…
    – Ethan Bolker
    10 hours ago










  • Thank you for the comment. However, my question is about the expected value that the algorithm will output m in list L.
    – Naseeb Thapaliya
    10 hours ago
















Perhaps this, or something like it: cs.stackexchange.com/questions/63682/…
– Ethan Bolker
10 hours ago




Perhaps this, or something like it: cs.stackexchange.com/questions/63682/…
– Ethan Bolker
10 hours ago












Thank you for the comment. However, my question is about the expected value that the algorithm will output m in list L.
– Naseeb Thapaliya
10 hours ago




Thank you for the comment. However, my question is about the expected value that the algorithm will output m in list L.
– Naseeb Thapaliya
10 hours ago










1 Answer
1






active

oldest

votes

















up vote
0
down vote













Let $X_i:=L[pi(i)]$, where $pi$ denotes the random permutation of $[n]$, $X_0equiv-infty$, and $M_i:=max_{jle i}X_j$. Then the expected number of resets is
begin{align}
&mathsf{E}left[sum_{i=1}^n 1{X_i>M_{i-1}}right]=sum_{i=1}^n mathsf{P}(X_i>M_{i-1}) \
&qquad =sum_{i=1}^nmathsf{E}left[mathsf{P}(X_i>M_{i-1}mid X_i)right]=frac{1}{n}sum_{i=1}^nsum_{j=1}^nbinom{j-1}{i-1}binom{n-1}{i-1}^{-1} \
&qquad=Psi(n+1)+gamma,
end{align}
where $Psi$ is the digamma function and $gamma=-Psi(1)$ is the Euler's constant.






share|cite|improve this answer























    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',
    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%2f3006710%2fexpected-number-of-updates-of-maximum%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








    up vote
    0
    down vote













    Let $X_i:=L[pi(i)]$, where $pi$ denotes the random permutation of $[n]$, $X_0equiv-infty$, and $M_i:=max_{jle i}X_j$. Then the expected number of resets is
    begin{align}
    &mathsf{E}left[sum_{i=1}^n 1{X_i>M_{i-1}}right]=sum_{i=1}^n mathsf{P}(X_i>M_{i-1}) \
    &qquad =sum_{i=1}^nmathsf{E}left[mathsf{P}(X_i>M_{i-1}mid X_i)right]=frac{1}{n}sum_{i=1}^nsum_{j=1}^nbinom{j-1}{i-1}binom{n-1}{i-1}^{-1} \
    &qquad=Psi(n+1)+gamma,
    end{align}
    where $Psi$ is the digamma function and $gamma=-Psi(1)$ is the Euler's constant.






    share|cite|improve this answer



























      up vote
      0
      down vote













      Let $X_i:=L[pi(i)]$, where $pi$ denotes the random permutation of $[n]$, $X_0equiv-infty$, and $M_i:=max_{jle i}X_j$. Then the expected number of resets is
      begin{align}
      &mathsf{E}left[sum_{i=1}^n 1{X_i>M_{i-1}}right]=sum_{i=1}^n mathsf{P}(X_i>M_{i-1}) \
      &qquad =sum_{i=1}^nmathsf{E}left[mathsf{P}(X_i>M_{i-1}mid X_i)right]=frac{1}{n}sum_{i=1}^nsum_{j=1}^nbinom{j-1}{i-1}binom{n-1}{i-1}^{-1} \
      &qquad=Psi(n+1)+gamma,
      end{align}
      where $Psi$ is the digamma function and $gamma=-Psi(1)$ is the Euler's constant.






      share|cite|improve this answer

























        up vote
        0
        down vote










        up vote
        0
        down vote









        Let $X_i:=L[pi(i)]$, where $pi$ denotes the random permutation of $[n]$, $X_0equiv-infty$, and $M_i:=max_{jle i}X_j$. Then the expected number of resets is
        begin{align}
        &mathsf{E}left[sum_{i=1}^n 1{X_i>M_{i-1}}right]=sum_{i=1}^n mathsf{P}(X_i>M_{i-1}) \
        &qquad =sum_{i=1}^nmathsf{E}left[mathsf{P}(X_i>M_{i-1}mid X_i)right]=frac{1}{n}sum_{i=1}^nsum_{j=1}^nbinom{j-1}{i-1}binom{n-1}{i-1}^{-1} \
        &qquad=Psi(n+1)+gamma,
        end{align}
        where $Psi$ is the digamma function and $gamma=-Psi(1)$ is the Euler's constant.






        share|cite|improve this answer














        Let $X_i:=L[pi(i)]$, where $pi$ denotes the random permutation of $[n]$, $X_0equiv-infty$, and $M_i:=max_{jle i}X_j$. Then the expected number of resets is
        begin{align}
        &mathsf{E}left[sum_{i=1}^n 1{X_i>M_{i-1}}right]=sum_{i=1}^n mathsf{P}(X_i>M_{i-1}) \
        &qquad =sum_{i=1}^nmathsf{E}left[mathsf{P}(X_i>M_{i-1}mid X_i)right]=frac{1}{n}sum_{i=1}^nsum_{j=1}^nbinom{j-1}{i-1}binom{n-1}{i-1}^{-1} \
        &qquad=Psi(n+1)+gamma,
        end{align}
        where $Psi$ is the digamma function and $gamma=-Psi(1)$ is the Euler's constant.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 1 hour ago

























        answered 2 hours ago









        d.k.o.

        8,079527




        8,079527






























             

            draft saved


            draft discarded



















































             


            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3006710%2fexpected-number-of-updates-of-maximum%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







            1G,qF9J38JClyINWPJH 5nx8vzHcmGcUqD6,0kjmWp,qE3raAtmSVMu
            -
            qiX13qxo,iMWKxQjAhoby,vopfu,TZZMr nSO Io3bhusdloJorF b,GGOPIcZ9bg7LgfWxmEPnSHDuiUelXXBtU xNnv

            Popular posts from this blog

            xlwings: Save and Close

            Image
            1 I am trying to find out how to save and close to an existing workbook using xlwings after writing in it: import xlwings as xw list_of_values = [1, 2, 3] workbook_path = 'abc.xlsx' wb = xw.Book(workbook_path) ws = wb.sheets['sheet1'] ws.range('E35').value = list_of_values wb.save() wb.close() When I get to wb.save(workbook_path) , there is a prompt stating: 'A file named abc.xlsx' already exists in this location. Do you want to replace it?' I want to overwrite the file immediately without the prompt coming up. According to the docs, wb.save() should automatically overwrite (see: https://docs.xlwings.org/en/v0.6.4/api.html). I have also tried wb.save(workbook_path) but the pop-up still appears. Appreciate any help on this please. p.s. - I am basically try...
            Read more

            UPSERT syntax error linked to UPDATE in PostgreSQL (python)

            Image
            1 1 I'm still learning PostgreSQL. During my testing, I have only been using INSERT statement in either psycopg2 and now asyncpg. I now have the need to UPDATE data in my test database, instead of replacing all of it. I'm currently trying to do a simple replacement test in a testing table, before I move to development table with more data. I want to replace any $1 name that is in CONFLICT with a name that is already in the table users. I'm trying the query code, which is passed to the DB via asyncpg. I keep getting a syntax errors, so I'm a little lost on how to correct these errors. What is the proper syntax for this query? '''INSERT INTO users(name, dob) VALUES($1, $2) ON CONFLICT (name) DO UPDATE "users" SET name = 'TEST' W...
            Read more

            Some classess of my CSS file are not rendering into Django templates (most classess render without problems)

            Image
            up vote 0 down vote favorite I'm starting with Django, I've been throught this several hours, please help me! on my *.html file I have this code (and more): {% load static %} <link rel="stylesheet" href="{% static 'css/style.css' %}"> and since I added that static file, my page has style.. but some classes don't render, for example this div: <div class="nav-profile-image"> If I delete the class .nav-profile-image from style.css, the page still the same, that image has no container, it's very strange. I uploaded all the app on branch "problems": https://github.com/franchodl/robohome/tree/problems view trying to render: home url for home: http://127.0.0.1:8000/propietario/ template: ./own/templates/own/home.html static files: ./static ...
            Read more