Heads and tails with unlimited capital












0












$begingroup$


Let $ S_{n} $ be the sum of won money by player 1 until moment $ n $

He gets 1 if he wins and loses 1 if he loses

Let $ X = infleft{ n : S_{n} = 1right} $

We assume that coin is asymetric and player 1 wins with probability $ p neq frac{1}{2}$

Calculate $ EX $.



Any hints or solutions would be appreciated






probability martingales stopping-times







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    0












    $begingroup$


    Let $ S_{n} $ be the sum of won money by player 1 until moment $ n $

    He gets 1 if he wins and loses 1 if he loses

    Let $ X = infleft{ n : S_{n} = 1right} $

    We assume that coin is asymetric and player 1 wins with probability $ p neq frac{1}{2}$

    Calculate $ EX $.



    Any hints or solutions would be appreciated






    probability martingales stopping-times







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Let $ S_{n} $ be the sum of won money by player 1 until moment $ n $

      He gets 1 if he wins and loses 1 if he loses

      Let $ X = infleft{ n : S_{n} = 1right} $

      We assume that coin is asymetric and player 1 wins with probability $ p neq frac{1}{2}$

      Calculate $ EX $.



      Any hints or solutions would be appreciated






      probability martingales stopping-times







      share|cite|improve this question









      $endgroup$




      Let $ S_{n} $ be the sum of won money by player 1 until moment $ n $

      He gets 1 if he wins and loses 1 if he loses

      Let $ X = infleft{ n : S_{n} = 1right} $

      We assume that coin is asymetric and player 1 wins with probability $ p neq frac{1}{2}$

      Calculate $ EX $.



      Any hints or solutions would be appreciated






      probability martingales stopping-times




      probability martingales stopping-times






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 9 at 10:27









      PabloPablo

      1456




      1456






















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          $begingroup$

          For $n=0,1,2,dots$:$$Pleft(X=2n+1right)=C_{n}p^{n+1}left(1-pright)^{n}$$
          where $C_n$ denotes the $n$-th Catalan number.



          So denoting $r:=p(1-p)$ we find:$$mathbb{E}X=psum_{n=0}^{infty}left(2n+1right)C_{n}r^{n}$$



          Setting $cleft(xright):=sum_{n=0}^{infty}C_{n}x^{n}$ we find:
          $$mathbb{E}X=pleft(2c'left(rright)r+cleft(rright)right)$$



          Here: $$cleft(xright)=frac{1-sqrt{1-4x}}{2x}=frac{2}{1+sqrt{1-4x}}$$ as you can find at the linked page.



          There might be a more simple way to find $mathbb EX$.






          share|cite|improve this answer











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            $begingroup$

            For $n=0,1,2,dots$:$$Pleft(X=2n+1right)=C_{n}p^{n+1}left(1-pright)^{n}$$
            where $C_n$ denotes the $n$-th Catalan number.



            So denoting $r:=p(1-p)$ we find:$$mathbb{E}X=psum_{n=0}^{infty}left(2n+1right)C_{n}r^{n}$$



            Setting $cleft(xright):=sum_{n=0}^{infty}C_{n}x^{n}$ we find:
            $$mathbb{E}X=pleft(2c'left(rright)r+cleft(rright)right)$$



            Here: $$cleft(xright)=frac{1-sqrt{1-4x}}{2x}=frac{2}{1+sqrt{1-4x}}$$ as you can find at the linked page.



            There might be a more simple way to find $mathbb EX$.






            share|cite|improve this answer











            $endgroup$


















              0












              $begingroup$

              For $n=0,1,2,dots$:$$Pleft(X=2n+1right)=C_{n}p^{n+1}left(1-pright)^{n}$$
              where $C_n$ denotes the $n$-th Catalan number.



              So denoting $r:=p(1-p)$ we find:$$mathbb{E}X=psum_{n=0}^{infty}left(2n+1right)C_{n}r^{n}$$



              Setting $cleft(xright):=sum_{n=0}^{infty}C_{n}x^{n}$ we find:
              $$mathbb{E}X=pleft(2c'left(rright)r+cleft(rright)right)$$



              Here: $$cleft(xright)=frac{1-sqrt{1-4x}}{2x}=frac{2}{1+sqrt{1-4x}}$$ as you can find at the linked page.



              There might be a more simple way to find $mathbb EX$.






              share|cite|improve this answer











              $endgroup$
















                0












                0








                0





                $begingroup$

                For $n=0,1,2,dots$:$$Pleft(X=2n+1right)=C_{n}p^{n+1}left(1-pright)^{n}$$
                where $C_n$ denotes the $n$-th Catalan number.



                So denoting $r:=p(1-p)$ we find:$$mathbb{E}X=psum_{n=0}^{infty}left(2n+1right)C_{n}r^{n}$$



                Setting $cleft(xright):=sum_{n=0}^{infty}C_{n}x^{n}$ we find:
                $$mathbb{E}X=pleft(2c'left(rright)r+cleft(rright)right)$$



                Here: $$cleft(xright)=frac{1-sqrt{1-4x}}{2x}=frac{2}{1+sqrt{1-4x}}$$ as you can find at the linked page.



                There might be a more simple way to find $mathbb EX$.






                share|cite|improve this answer











                $endgroup$



                For $n=0,1,2,dots$:$$Pleft(X=2n+1right)=C_{n}p^{n+1}left(1-pright)^{n}$$
                where $C_n$ denotes the $n$-th Catalan number.



                So denoting $r:=p(1-p)$ we find:$$mathbb{E}X=psum_{n=0}^{infty}left(2n+1right)C_{n}r^{n}$$



                Setting $cleft(xright):=sum_{n=0}^{infty}C_{n}x^{n}$ we find:
                $$mathbb{E}X=pleft(2c'left(rright)r+cleft(rright)right)$$



                Here: $$cleft(xright)=frac{1-sqrt{1-4x}}{2x}=frac{2}{1+sqrt{1-4x}}$$ as you can find at the linked page.



                There might be a more simple way to find $mathbb EX$.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Jan 9 at 11:33

























                answered Jan 9 at 11:23









                drhabdrhab

                104k545136




                104k545136






























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