Suppose that X1,...,Xn form a random sample from a Poisson

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Let random variables  have joint pdf or pmb

where the parameters  are unknown. When function  is a function of parameters  , it is called the

likelihood function

. Values  that maximize the likelihood function are the maximum likelihood estimates (mle's), or equally values  for which

for every . By substituting  with , the

maximum likelihood estimators

are obtained.

A random variable  with pmf

for , is said to have Poisson Distribution with parameter .