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stochastic domination of binomial distributions

0

$begingroup$

Consider the process of adding $k$ balls into a box:

• each ball $b_i$ will be added with its own probability $P(b_i in B)$
  


• for the first $k' = lfloor rho k rfloor $ balls we know that $P(b_i in B) geq gamma$
  

now clearly the random variable $|B|$ of balls inside the box dominates a $Bin(k',gamma)$ distributed random variable $X$
i.e. $P(|B|geq x) geq P(X geq x)$

now I want to show, that there exist $delta > 0$ and $p<1$ only depending on $rho$ and $gamma$ such that

$P(|B| leq delta k ) leq p^k$

Any Advice would be greatly appreciated

probability-theory binomial-distribution

share|cite|improve this question

asked Mar 20 at 11:22

user373499user373499

205

$endgroup$

add a comment | 

0

$begingroup$

Consider the process of adding $k$ balls into a box:

• each ball $b_i$ will be added with its own probability $P(b_i in B)$
  


• for the first $k' = lfloor rho k rfloor $ balls we know that $P(b_i in B) geq gamma$
  

now clearly the random variable $|B|$ of balls inside the box dominates a $Bin(k',gamma)$ distributed random variable $X$
i.e. $P(|B|geq x) geq P(X geq x)$

now I want to show, that there exist $delta > 0$ and $p<1$ only depending on $rho$ and $gamma$ such that

$P(|B| leq delta k ) leq p^k$

Any Advice would be greatly appreciated

probability-theory binomial-distribution

share|cite|improve this question

asked Mar 20 at 11:22

user373499user373499

205

$endgroup$

add a comment | 

$begingroup$

Consider the process of adding $k$ balls into a box:

• each ball $b_i$ will be added with its own probability $P(b_i in B)$
  


• for the first $k' = lfloor rho k rfloor $ balls we know that $P(b_i in B) geq gamma$
  

now clearly the random variable $|B|$ of balls inside the box dominates a $Bin(k',gamma)$ distributed random variable $X$
i.e. $P(|B|geq x) geq P(X geq x)$

now I want to show, that there exist $delta > 0$ and $p<1$ only depending on $rho$ and $gamma$ such that

$P(|B| leq delta k ) leq p^k$

Any Advice would be greatly appreciated

probability-theory binomial-distribution

share|cite|improve this question

asked Mar 20 at 11:22

user373499user373499

205

$endgroup$

share|cite|improve this question

asked Mar 20 at 11:22

user373499user373499

205

share|cite|improve this question

asked Mar 20 at 11:22

user373499user373499

205

asked Mar 20 at 11:22

user373499user373499

205

asked Mar 20 at 11:22

user373499user373499

205