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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 $n in mathbb{N}$ 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: begin{align*}q_0 =& 1-p_0\q_1 =& 1-q_0end{align*}
  


• If $n=2$, the distribution is: begin{align*}q_0=&(p_0-1)^2\ q_1=&(p_0-1)p_0(p_1-1)\ q_2 =& 1-q_0-q_1end{align*}
  


• If $n=3$, the distribution is: begin{align*}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 end{align*}
  

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: $l_1 = sqrt{2}, l_2=frac{1}{sqrt{3}},l_3 = frac{27}{280},l_4=frac{73162}{5010005sqrt{5}}$.

probability probability-distributions

share|cite|improve this question

edited Mar 20 at 19:57

J. W. Tanner

4,6191320

asked Mar 20 at 19:37

user1747134user1747134

239

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  

  
  
  
  

add a comment | 

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 $n in mathbb{N}$ 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: begin{align*}q_0 =& 1-p_0\q_1 =& 1-q_0end{align*}
  


• If $n=2$, the distribution is: begin{align*}q_0=&(p_0-1)^2\ q_1=&(p_0-1)p_0(p_1-1)\ q_2 =& 1-q_0-q_1end{align*}
  


• If $n=3$, the distribution is: begin{align*}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 end{align*}
  

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: $l_1 = sqrt{2}, l_2=frac{1}{sqrt{3}},l_3 = frac{27}{280},l_4=frac{73162}{5010005sqrt{5}}$.

probability probability-distributions

share|cite|improve this question

edited Mar 20 at 19:57

J. W. Tanner

4,6191320

asked Mar 20 at 19:37

user1747134user1747134

239

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  

  
  
  
  

add a comment | 

$begingroup$

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 $n in mathbb{N}$ 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: begin{align*}q_0 =& 1-p_0\q_1 =& 1-q_0end{align*}
  


• If $n=2$, the distribution is: begin{align*}q_0=&(p_0-1)^2\ q_1=&(p_0-1)p_0(p_1-1)\ q_2 =& 1-q_0-q_1end{align*}
  


• If $n=3$, the distribution is: begin{align*}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 end{align*}
  

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: $l_1 = sqrt{2}, l_2=frac{1}{sqrt{3}},l_3 = frac{27}{280},l_4=frac{73162}{5010005sqrt{5}}$.

probability probability-distributions

share|cite|improve this question

edited Mar 20 at 19:57

J. W. Tanner

4,6191320

asked Mar 20 at 19:37

user1747134user1747134

239

$endgroup$

share|cite|improve this question

edited Mar 20 at 19:57

J. W. Tanner

4,6191320

asked Mar 20 at 19:37

user1747134user1747134

239

share|cite|improve this question

edited Mar 20 at 19:57

J. W. Tanner

4,6191320

asked Mar 20 at 19:37

user1747134user1747134

239

edited Mar 20 at 19:57

J. W. Tanner

4,6191320

edited Mar 20 at 19:57

J. W. Tanner

4,6191320

asked Mar 20 at 19:37

user1747134user1747134

239

asked Mar 20 at 19:37

user1747134user1747134

239