Consider the sample x_(1),dots,x_(n)∼^(iid)Poisson(\lambda ) for some \lambda >0, and let \eta =e^(-\lambda ). For each i, let Y_(i)=1 if x_(i)=0,
and Y_(i)=0 if x_(i)>1.
(a) What is the distribution of Y_(i), for each i ? Identify this distribution by name.
(b) One estimator for \eta is ()/(bar) (Y)=(1)/(n)\sum_(i=1)^n Y_(i). Compute the bias and the mean squared error for this estimator.
Does this estimator improve as the sample size grows?