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Example of Cherenkov
Additivity Property of Binomial or Poisson Distributions
These two distributions (Binomial and Poisson) share an important additivity property, which is
, obvious and
, leads to confusion when used in conjunction with central limit theorem (see later).
(a) Suppose the independent random variables
and
have binomial distributions with parameters
and
respectively.
Here,
if there are
k
successes.
Then it is obvious from source of binomial distribution that
has a binomial distribution with parameter
.
(b) Suppose independent random variables
and
have Poisson distributions
and
.
Then
has Poisson distribution with parameter
.
This is obvious either algebraically, or by using
the result (a) above with
and
and let
,
, fixed
,
.
Geoffrey Fox
,
Northeast Parallel Architectures Center
at Syracuse University,
gcf@npac.syr.edu