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Looking for a generalization of Binomial distribution and its properties


Binomial Distribution Parameter & ProbabilityOrder statistics of a max-min problem.Posterior for Beta Binomial Distribution with Repeated ObservationsThe Marginal Distribution of a Multinomialhow far the distribution from the uniform distributionDifferences between Binomial and Normal Distribution ModelsSucces and failure probability for dependant trials.Multiple conditions for Bayes Theorem to extract multivariate posterior distributionIs there a generalization of the concept of variance for a collection of probability distributions?Identifying a distribution by its properties













0












$begingroup$


I asked this on MathOverflow, but was redirected here. Anyway:



In my research (coming from computer science), I have encountered a family of discrete probability distributions that seems to be some sort of generalization of the binomial distribution. A distribution in this family has a parameter $n in mathbbN$ and parameters $p_0,... ,p_n-1 in [0,1]$. The distribution has $n+1$ possible values. Denote the respective probablities $q_0,...,q_n$. So far, I have managed to obtain the descriptions for the first few $n$, which are the following:



  • If $n=1$, the distribution is: beginalign*q_0 =& 1-p_0\q_1 =& 1-q_0endalign*

  • If $n=2$, the distribution is: beginalign*q_0=&(p_0-1)^2\ q_1=&(p_0-1)p_0(p_1-1)\ q_2 =& 1-q_0-q_1endalign*

  • If $n=3$, the distribution is: beginalign*q_0=&(1 - p_0)^3\q_1 =& 3 (p_0-1)^2 p_0 (p_1-1)^2 \ q_2=& -3 (p_0-1) p_0 (p_1-1)^2 (2 (p_0-1) p_1-p_0) (p_2-1)\q_3 =& 1-q_0-q_1-q_2 endalign*

I could give the descriptions for few higher $n$, but it becomes lots of symbols quickly.



I have noticed, that when I set $p_i = 0 ; forall i = 1,...,n-1$ it degenerates to the distribution $Bin(n,p_0)$. I have several goals with this. First, I wanted to ask if this resembles some well known distribution family generalizing the binomial distribution? Can anyone suggest possible general formulas that would produce $q_i$ for a given $n$? Assuming, we have a generalization, what would be the measure of all the distributions representable as a member of this family for given $n$ or some upper bound of such measure? Denoting this measure $l_n$, I have computed that: $l_1 = sqrt2, l_2=frac1sqrt3,l_3 = frac27280,l_4=frac731625010005sqrt5$.










share|cite|improve this question











$endgroup$











  • $begingroup$
    Can you describe exactly the sampling process you are referring to? Right now, you haven't, and we are left to figure out how the distribution you're considering is even defined.
    $endgroup$
    – Clement C.
    Mar 20 at 20:51










  • $begingroup$
    It's somewhat complicated and I didn't want to go into the details. I was mostly just hoping this could resemble something well known to give me some hints where to look.
    $endgroup$
    – user1747134
    Mar 21 at 9:57










  • $begingroup$
    Without those details, this is more like a shot in the dark than anything else....
    $endgroup$
    – Clement C.
    Mar 21 at 16:25















0












$begingroup$


I asked this on MathOverflow, but was redirected here. Anyway:



In my research (coming from computer science), I have encountered a family of discrete probability distributions that seems to be some sort of generalization of the binomial distribution. A distribution in this family has a parameter $n in mathbbN$ and parameters $p_0,... ,p_n-1 in [0,1]$. The distribution has $n+1$ possible values. Denote the respective probablities $q_0,...,q_n$. So far, I have managed to obtain the descriptions for the first few $n$, which are the following:



  • If $n=1$, the distribution is: beginalign*q_0 =& 1-p_0\q_1 =& 1-q_0endalign*

  • If $n=2$, the distribution is: beginalign*q_0=&(p_0-1)^2\ q_1=&(p_0-1)p_0(p_1-1)\ q_2 =& 1-q_0-q_1endalign*

  • If $n=3$, the distribution is: beginalign*q_0=&(1 - p_0)^3\q_1 =& 3 (p_0-1)^2 p_0 (p_1-1)^2 \ q_2=& -3 (p_0-1) p_0 (p_1-1)^2 (2 (p_0-1) p_1-p_0) (p_2-1)\q_3 =& 1-q_0-q_1-q_2 endalign*

I could give the descriptions for few higher $n$, but it becomes lots of symbols quickly.



I have noticed, that when I set $p_i = 0 ; forall i = 1,...,n-1$ it degenerates to the distribution $Bin(n,p_0)$. I have several goals with this. First, I wanted to ask if this resembles some well known distribution family generalizing the binomial distribution? Can anyone suggest possible general formulas that would produce $q_i$ for a given $n$? Assuming, we have a generalization, what would be the measure of all the distributions representable as a member of this family for given $n$ or some upper bound of such measure? Denoting this measure $l_n$, I have computed that: $l_1 = sqrt2, l_2=frac1sqrt3,l_3 = frac27280,l_4=frac731625010005sqrt5$.










share|cite|improve this question











$endgroup$











  • $begingroup$
    Can you describe exactly the sampling process you are referring to? Right now, you haven't, and we are left to figure out how the distribution you're considering is even defined.
    $endgroup$
    – Clement C.
    Mar 20 at 20:51










  • $begingroup$
    It's somewhat complicated and I didn't want to go into the details. I was mostly just hoping this could resemble something well known to give me some hints where to look.
    $endgroup$
    – user1747134
    Mar 21 at 9:57










  • $begingroup$
    Without those details, this is more like a shot in the dark than anything else....
    $endgroup$
    – Clement C.
    Mar 21 at 16:25













0












0








0





$begingroup$


I asked this on MathOverflow, but was redirected here. Anyway:



In my research (coming from computer science), I have encountered a family of discrete probability distributions that seems to be some sort of generalization of the binomial distribution. A distribution in this family has a parameter $n in mathbbN$ and parameters $p_0,... ,p_n-1 in [0,1]$. The distribution has $n+1$ possible values. Denote the respective probablities $q_0,...,q_n$. So far, I have managed to obtain the descriptions for the first few $n$, which are the following:



  • If $n=1$, the distribution is: beginalign*q_0 =& 1-p_0\q_1 =& 1-q_0endalign*

  • If $n=2$, the distribution is: beginalign*q_0=&(p_0-1)^2\ q_1=&(p_0-1)p_0(p_1-1)\ q_2 =& 1-q_0-q_1endalign*

  • If $n=3$, the distribution is: beginalign*q_0=&(1 - p_0)^3\q_1 =& 3 (p_0-1)^2 p_0 (p_1-1)^2 \ q_2=& -3 (p_0-1) p_0 (p_1-1)^2 (2 (p_0-1) p_1-p_0) (p_2-1)\q_3 =& 1-q_0-q_1-q_2 endalign*

I could give the descriptions for few higher $n$, but it becomes lots of symbols quickly.



I have noticed, that when I set $p_i = 0 ; forall i = 1,...,n-1$ it degenerates to the distribution $Bin(n,p_0)$. I have several goals with this. First, I wanted to ask if this resembles some well known distribution family generalizing the binomial distribution? Can anyone suggest possible general formulas that would produce $q_i$ for a given $n$? Assuming, we have a generalization, what would be the measure of all the distributions representable as a member of this family for given $n$ or some upper bound of such measure? Denoting this measure $l_n$, I have computed that: $l_1 = sqrt2, l_2=frac1sqrt3,l_3 = frac27280,l_4=frac731625010005sqrt5$.










share|cite|improve this question











$endgroup$




I asked this on MathOverflow, but was redirected here. Anyway:



In my research (coming from computer science), I have encountered a family of discrete probability distributions that seems to be some sort of generalization of the binomial distribution. A distribution in this family has a parameter $n in mathbbN$ and parameters $p_0,... ,p_n-1 in [0,1]$. The distribution has $n+1$ possible values. Denote the respective probablities $q_0,...,q_n$. So far, I have managed to obtain the descriptions for the first few $n$, which are the following:



  • If $n=1$, the distribution is: beginalign*q_0 =& 1-p_0\q_1 =& 1-q_0endalign*

  • If $n=2$, the distribution is: beginalign*q_0=&(p_0-1)^2\ q_1=&(p_0-1)p_0(p_1-1)\ q_2 =& 1-q_0-q_1endalign*

  • If $n=3$, the distribution is: beginalign*q_0=&(1 - p_0)^3\q_1 =& 3 (p_0-1)^2 p_0 (p_1-1)^2 \ q_2=& -3 (p_0-1) p_0 (p_1-1)^2 (2 (p_0-1) p_1-p_0) (p_2-1)\q_3 =& 1-q_0-q_1-q_2 endalign*

I could give the descriptions for few higher $n$, but it becomes lots of symbols quickly.



I have noticed, that when I set $p_i = 0 ; forall i = 1,...,n-1$ it degenerates to the distribution $Bin(n,p_0)$. I have several goals with this. First, I wanted to ask if this resembles some well known distribution family generalizing the binomial distribution? Can anyone suggest possible general formulas that would produce $q_i$ for a given $n$? Assuming, we have a generalization, what would be the measure of all the distributions representable as a member of this family for given $n$ or some upper bound of such measure? Denoting this measure $l_n$, I have computed that: $l_1 = sqrt2, l_2=frac1sqrt3,l_3 = frac27280,l_4=frac731625010005sqrt5$.







probability probability-distributions






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 20 at 19:57









J. W. Tanner

4,2661320




4,2661320










asked Mar 20 at 19:37









user1747134user1747134

219




219











  • $begingroup$
    Can you describe exactly the sampling process you are referring to? Right now, you haven't, and we are left to figure out how the distribution you're considering is even defined.
    $endgroup$
    – Clement C.
    Mar 20 at 20:51










  • $begingroup$
    It's somewhat complicated and I didn't want to go into the details. I was mostly just hoping this could resemble something well known to give me some hints where to look.
    $endgroup$
    – user1747134
    Mar 21 at 9:57










  • $begingroup$
    Without those details, this is more like a shot in the dark than anything else....
    $endgroup$
    – Clement C.
    Mar 21 at 16:25
















  • $begingroup$
    Can you describe exactly the sampling process you are referring to? Right now, you haven't, and we are left to figure out how the distribution you're considering is even defined.
    $endgroup$
    – Clement C.
    Mar 20 at 20:51










  • $begingroup$
    It's somewhat complicated and I didn't want to go into the details. I was mostly just hoping this could resemble something well known to give me some hints where to look.
    $endgroup$
    – user1747134
    Mar 21 at 9:57










  • $begingroup$
    Without those details, this is more like a shot in the dark than anything else....
    $endgroup$
    – Clement C.
    Mar 21 at 16:25















$begingroup$
Can you describe exactly the sampling process you are referring to? Right now, you haven't, and we are left to figure out how the distribution you're considering is even defined.
$endgroup$
– Clement C.
Mar 20 at 20:51




$begingroup$
Can you describe exactly the sampling process you are referring to? Right now, you haven't, and we are left to figure out how the distribution you're considering is even defined.
$endgroup$
– Clement C.
Mar 20 at 20:51












$begingroup$
It's somewhat complicated and I didn't want to go into the details. I was mostly just hoping this could resemble something well known to give me some hints where to look.
$endgroup$
– user1747134
Mar 21 at 9:57




$begingroup$
It's somewhat complicated and I didn't want to go into the details. I was mostly just hoping this could resemble something well known to give me some hints where to look.
$endgroup$
– user1747134
Mar 21 at 9:57












$begingroup$
Without those details, this is more like a shot in the dark than anything else....
$endgroup$
– Clement C.
Mar 21 at 16:25




$begingroup$
Without those details, this is more like a shot in the dark than anything else....
$endgroup$
– Clement C.
Mar 21 at 16:25










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