The number N of species of insect caught in a trap during one night in a certain region is modelled by a distribution of the form P(N=n)=(p^(n))/(-n log(1-p)) for n=1,2,3,dots, where the unknown parameter p must lie between 0 and 1 . Forty independent observations N_(1),N_(2),dots,N_(40) are made. (a) Show that the mean of this distribution is E(N)=-p((1-p)log(1-p))^(-1) (b) Find an equation that determines the maximum likelihood estimator of p. (Do not attempt to solve this equation.) (c) Using a large sample property of maximum likelihood estimators, find the approximate distribution of hat(p) when n is large.
1. The number N of species of insect caught in a trap during one night in a certain region is modelled by a distribution of the form
pn
for n = 1,2,3,.. . , where the unknown parameter p must lie between 0 and 1. Forty indepen dent observations N1,N2,... N4o are made.
(a) Show that the mean of this distribution is
E(N)= -p((1 - p) log(1 - p))-l
(b) Find an equation that determines the maximum likelihood estimator of p. (Do not attempt to solve this equation.)
(c) Using a large sample property of maximum likelihood estimators, find the approximate distribution of p when n is large.