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In bacterial counts with a haemacytometer, the number of bacteria per quadrat has a Pois- son distribution with probability mass function f(x), where f(x) = 0Te-® /x! and 0 is to be estimated. If there are many bacteria in a quadrat, it is difficult to count them all, and so the only information recorded is that the number of bacteria exceeds a certain limit c, a large positive integer. In a random sample of n quadrats, it was observed that ng quadrats had x bacteria (x = 0,1, · ·,c) and that m quadrats had more than c bacteria. Show that the maximum likelihood estimate of 0 is given by 1 mf(c) = s In - m - where s = an, and F (c) Σf(a). x=1 x=0

In bacterial counts with a haemacytometer, the number of bacteria per quadrat has a Pois- son distribution with probability mass function f(x), where f(x) = 0Te-® /x! and 0 is to be estimated. If there are many bacteria in a quadrat, it is difficult to count them all, and so the only information recorded is that the number of bacteria exceeds a certain limit c, a large positive integer. In a random sample of n quadrats, it was observed that ng quadrats had x bacteria (x = 0,1, · ·,c) and that m quadrats had more than c bacteria. Show that the maximum likelihood estimate of 0 is given by 1 mf(c) = s In - m - where s = an, and F (c) Σf(a). x=1 x=0

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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7. In bacterial counts with a haemacytometer, the number of bacteria per quadrat has a Pois- son distribution with probability mass function f(x), where f(x) = 0" e- /x! and 0 is to be estimated. If there are many bacteria in a quadrat, it is difficult to count them all, and so the only information recorded is that the number of bacteria exceeds a certain limit c, a large positive integer. In a random sample of n quadrats, it was observed that ng quadrats had x bacteria (x = 0,1, ., c) and that m quadrats had more than c bacteria. Show that the maximum likelihood estimate of 0 is given by -1 mf(c) 1- F(c)] = s In – m - where s = Ecn, and F(c) =E{(x). x=1 x=0
Transcribed Image Text:7. In bacterial counts with a haemacytometer, the number of bacteria per quadrat has a Pois- son distribution with probability mass function f(x), where f(x) = 0" e- /x! and 0 is to be estimated. If there are many bacteria in a quadrat, it is difficult to count them all, and so the only information recorded is that the number of bacteria exceeds a certain limit c, a large positive integer. In a random sample of n quadrats, it was observed that ng quadrats had x bacteria (x = 0,1, ., c) and that m quadrats had more than c bacteria. Show that the maximum likelihood estimate of 0 is given by -1 mf(c) 1- F(c)] = s In – m - where s = Ecn, and F(c) =E{(x). x=1 x=0
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