Generating function of random variable $X$

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Suppose random variable $X$ with values $2,3,...$ has distribution $P(X=k) = (k-1)p^2(1-p)^k-2$. Show that $$G_x(s) = left(fracps1-s(1-p)right)^2$$
Is this easy to show by definition or how do I solve it?







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    Suppose random variable $X$ with values $2,3,...$ has distribution $P(X=k) = (k-1)p^2(1-p)^k-2$. Show that $$G_x(s) = left(fracps1-s(1-p)right)^2$$
    Is this easy to show by definition or how do I solve it?







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      Suppose random variable $X$ with values $2,3,...$ has distribution $P(X=k) = (k-1)p^2(1-p)^k-2$. Show that $$G_x(s) = left(fracps1-s(1-p)right)^2$$
      Is this easy to show by definition or how do I solve it?







      share|cite|improve this question












      Suppose random variable $X$ with values $2,3,...$ has distribution $P(X=k) = (k-1)p^2(1-p)^k-2$. Show that $$G_x(s) = left(fracps1-s(1-p)right)^2$$
      Is this easy to show by definition or how do I solve it?









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      asked Aug 20 at 9:14









      mandella

      713420




      713420




















          2 Answers
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          Fot $0<t<1$, $sum_k=0^infty t^k= frac 1 1-t$ and $sum_k=0^infty kt^k-1= -frac 1 (1-t)^2$. This gives $sum_k=0^infty kt^k= -frac t (1-t)^2$. Now write down $sum t^kPX=k$ and use these formulas to compute $G_X(t)$.






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          • got it thank you
            – mandella
            Aug 20 at 9:29

















          up vote
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          Hint Remember the definition of $G_X$ :



          $$ G_X(s) = sum_n=0^infty mathbb P (X=k)s^k $$



          which is defined on $[-1,1]$. So with that definition, and by making appear a geometric sum (or something close to it) you might be able to find the answer.






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            2 Answers
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            active

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            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            1
            down vote













            Fot $0<t<1$, $sum_k=0^infty t^k= frac 1 1-t$ and $sum_k=0^infty kt^k-1= -frac 1 (1-t)^2$. This gives $sum_k=0^infty kt^k= -frac t (1-t)^2$. Now write down $sum t^kPX=k$ and use these formulas to compute $G_X(t)$.






            share|cite|improve this answer




















            • got it thank you
              – mandella
              Aug 20 at 9:29














            up vote
            1
            down vote













            Fot $0<t<1$, $sum_k=0^infty t^k= frac 1 1-t$ and $sum_k=0^infty kt^k-1= -frac 1 (1-t)^2$. This gives $sum_k=0^infty kt^k= -frac t (1-t)^2$. Now write down $sum t^kPX=k$ and use these formulas to compute $G_X(t)$.






            share|cite|improve this answer




















            • got it thank you
              – mandella
              Aug 20 at 9:29












            up vote
            1
            down vote










            up vote
            1
            down vote









            Fot $0<t<1$, $sum_k=0^infty t^k= frac 1 1-t$ and $sum_k=0^infty kt^k-1= -frac 1 (1-t)^2$. This gives $sum_k=0^infty kt^k= -frac t (1-t)^2$. Now write down $sum t^kPX=k$ and use these formulas to compute $G_X(t)$.






            share|cite|improve this answer












            Fot $0<t<1$, $sum_k=0^infty t^k= frac 1 1-t$ and $sum_k=0^infty kt^k-1= -frac 1 (1-t)^2$. This gives $sum_k=0^infty kt^k= -frac t (1-t)^2$. Now write down $sum t^kPX=k$ and use these formulas to compute $G_X(t)$.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Aug 20 at 9:26









            Kavi Rama Murthy

            23.3k2933




            23.3k2933











            • got it thank you
              – mandella
              Aug 20 at 9:29
















            • got it thank you
              – mandella
              Aug 20 at 9:29















            got it thank you
            – mandella
            Aug 20 at 9:29




            got it thank you
            – mandella
            Aug 20 at 9:29










            up vote
            0
            down vote













            Hint Remember the definition of $G_X$ :



            $$ G_X(s) = sum_n=0^infty mathbb P (X=k)s^k $$



            which is defined on $[-1,1]$. So with that definition, and by making appear a geometric sum (or something close to it) you might be able to find the answer.






            share|cite|improve this answer
























              up vote
              0
              down vote













              Hint Remember the definition of $G_X$ :



              $$ G_X(s) = sum_n=0^infty mathbb P (X=k)s^k $$



              which is defined on $[-1,1]$. So with that definition, and by making appear a geometric sum (or something close to it) you might be able to find the answer.






              share|cite|improve this answer






















                up vote
                0
                down vote










                up vote
                0
                down vote









                Hint Remember the definition of $G_X$ :



                $$ G_X(s) = sum_n=0^infty mathbb P (X=k)s^k $$



                which is defined on $[-1,1]$. So with that definition, and by making appear a geometric sum (or something close to it) you might be able to find the answer.






                share|cite|improve this answer












                Hint Remember the definition of $G_X$ :



                $$ G_X(s) = sum_n=0^infty mathbb P (X=k)s^k $$



                which is defined on $[-1,1]$. So with that definition, and by making appear a geometric sum (or something close to it) you might be able to find the answer.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Aug 20 at 9:23









                tmaths

                1,326113




                1,326113






















                     

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