Infinite coin flipping Show $ sum_{k=0}^{infty} f(k,r,p) = 1 $












2














For a infinite coin flip consider the probability of success $ p in (0,1) $.



For $n in mathbb{N}_0 $ and $ r in mathbb{N} $ define the probability $f(k,r,p)$ ( $k$ failures then the $r$th sucess ).



Show:



$ sum_{k=0}^{infty} f(k,r,p) = 1 $.



So I've already have a possible solution but I want to know if my solution is ok. So can you please check my attempt?



I want to ask a tiny question before I try to prove the equation: $f$ is a generalization of a geometric distribution, right? Anyways:



So $ sum_{k=0}^{infty} f(k,r,p) = sum_{k=0}^{infty} {k+r-1 choose k}p^r(1-p)^k = p^r sum_{k=0}^{infty} {k+r-1 choose k}(-1)^k(-(1-p))^k =p^r sum_{k=0}^{infty}{-r choose k}(-(1-p))^k = p^r(1-(1-p))^{-r} = p^rp^{-r} = 1.$



What did I do in the third equation? : ${k+r-1 choose k}(-1)^k = {-r choose k} $ .










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  • 1




    Your argument is correct. About your tiny question, the distribution is known as Negative Binomial
    – Daniel
    Nov 21 at 22:14
















2














For a infinite coin flip consider the probability of success $ p in (0,1) $.



For $n in mathbb{N}_0 $ and $ r in mathbb{N} $ define the probability $f(k,r,p)$ ( $k$ failures then the $r$th sucess ).



Show:



$ sum_{k=0}^{infty} f(k,r,p) = 1 $.



So I've already have a possible solution but I want to know if my solution is ok. So can you please check my attempt?



I want to ask a tiny question before I try to prove the equation: $f$ is a generalization of a geometric distribution, right? Anyways:



So $ sum_{k=0}^{infty} f(k,r,p) = sum_{k=0}^{infty} {k+r-1 choose k}p^r(1-p)^k = p^r sum_{k=0}^{infty} {k+r-1 choose k}(-1)^k(-(1-p))^k =p^r sum_{k=0}^{infty}{-r choose k}(-(1-p))^k = p^r(1-(1-p))^{-r} = p^rp^{-r} = 1.$



What did I do in the third equation? : ${k+r-1 choose k}(-1)^k = {-r choose k} $ .










share|cite|improve this question




















  • 1




    Your argument is correct. About your tiny question, the distribution is known as Negative Binomial
    – Daniel
    Nov 21 at 22:14














2












2








2


1





For a infinite coin flip consider the probability of success $ p in (0,1) $.



For $n in mathbb{N}_0 $ and $ r in mathbb{N} $ define the probability $f(k,r,p)$ ( $k$ failures then the $r$th sucess ).



Show:



$ sum_{k=0}^{infty} f(k,r,p) = 1 $.



So I've already have a possible solution but I want to know if my solution is ok. So can you please check my attempt?



I want to ask a tiny question before I try to prove the equation: $f$ is a generalization of a geometric distribution, right? Anyways:



So $ sum_{k=0}^{infty} f(k,r,p) = sum_{k=0}^{infty} {k+r-1 choose k}p^r(1-p)^k = p^r sum_{k=0}^{infty} {k+r-1 choose k}(-1)^k(-(1-p))^k =p^r sum_{k=0}^{infty}{-r choose k}(-(1-p))^k = p^r(1-(1-p))^{-r} = p^rp^{-r} = 1.$



What did I do in the third equation? : ${k+r-1 choose k}(-1)^k = {-r choose k} $ .










share|cite|improve this question















For a infinite coin flip consider the probability of success $ p in (0,1) $.



For $n in mathbb{N}_0 $ and $ r in mathbb{N} $ define the probability $f(k,r,p)$ ( $k$ failures then the $r$th sucess ).



Show:



$ sum_{k=0}^{infty} f(k,r,p) = 1 $.



So I've already have a possible solution but I want to know if my solution is ok. So can you please check my attempt?



I want to ask a tiny question before I try to prove the equation: $f$ is a generalization of a geometric distribution, right? Anyways:



So $ sum_{k=0}^{infty} f(k,r,p) = sum_{k=0}^{infty} {k+r-1 choose k}p^r(1-p)^k = p^r sum_{k=0}^{infty} {k+r-1 choose k}(-1)^k(-(1-p))^k =p^r sum_{k=0}^{infty}{-r choose k}(-(1-p))^k = p^r(1-(1-p))^{-r} = p^rp^{-r} = 1.$



What did I do in the third equation? : ${k+r-1 choose k}(-1)^k = {-r choose k} $ .







calculus sequences-and-series probability-theory probability-distributions






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edited Nov 21 at 22:06

























asked Nov 21 at 21:26









RukiaKuchiki

299211




299211








  • 1




    Your argument is correct. About your tiny question, the distribution is known as Negative Binomial
    – Daniel
    Nov 21 at 22:14














  • 1




    Your argument is correct. About your tiny question, the distribution is known as Negative Binomial
    – Daniel
    Nov 21 at 22:14








1




1




Your argument is correct. About your tiny question, the distribution is known as Negative Binomial
– Daniel
Nov 21 at 22:14




Your argument is correct. About your tiny question, the distribution is known as Negative Binomial
– Daniel
Nov 21 at 22:14















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