The Poisson binomial distribution is the statistical distribution of a sum
 |
(1)
|
of independent random variables 
having Bernoulli distributions
with possibly different success probabilities 
. Its probability generating
function is
![G_X(z)=product_(i=1)^n[1-p_i+p_iz].](/images/equations/PoissonBinomialDistribution/NumberedEquation2.svg) |
(2)
|
Consequently, 
is the coefficient of 
in this product, while the expectation
value and variance are 
and 
, respectively. If every 
is equal, the distribution reduces to the binomial
distribution. When the probabilities 
are small, a Poisson
distribution with mean 
can provide an approximation.
For integers 
and 
, let 
be the infimum of means
of finite sums 
of independent
random variables having Bernoulli distributions
for which 
.
Kleiner (2026) proved that 
and 
for 
. For distance two, 
, where
 |
(3)
|
and 
,
with 
the floor function. In particular, 
as 
, where 
is big-O notation.
