Person P is walking with his dog every day. He decides to model the time intervals between dog encounters (i.e., how long does it take between two encounters) with the independent random variables Y₁, . . ., Yn . Number of dogs encounters per day, the number of customers entering a store per hour, the number of bacteria per liter of water, etc. are classically modelled using the Poisson distribution. With an exponential distribution, on the other hand, is used for modelling intervals of between events and how long a light bulb lasts, for example. After learning this, person P decides to model the observations in such a way that the corresponding random variables Y1,..., Yn for a random sample, that is, independent observation, of the exponential distribution Exp(1/µ), µ > 0. The parameter µ > 0 is used as the statistical model parameter 1 which is the expected value of each observation random variable Yi. (a) Form the maximum likelihood estimate μ^(y) in the model for the parameter μ and form the maximum likelihood estimator µ^(Y) in the model for the parameter μ. (b) Assume that n = intervals2 (min)) is 6 and the observed data (i.e., six dog encounter y=(y1,y2,...,y5,y6)=(2.25, 8.40, 5.10, 11.85, 3.30, 8.10). Use the data to calculate the ML estimate for the expected value μ of the intervals. Also consider what can be deduced from this: On the basis on this analysis of problem, do you think it is possible, i.e. likely, that expected value would actually be 3 (minutes). There is no "right answer" to this.
Person P is walking with his dog every day. He decides to model the time intervals between dog encounters (i.e., how long does it take between two encounters) with the independent random variables Y₁, . . ., Yn . Number of dogs encounters per day, the number of customers entering a store per hour, the number of bacteria per liter of water, etc. are classically modelled using the Poisson distribution. With an exponential distribution, on the other hand, is used for modelling intervals of between events and how long a light bulb lasts, for example. After learning this, person P decides to model the observations in such a way that the corresponding random variables Y1,..., Yn for a random sample, that is, independent observation, of the exponential distribution Exp(1/µ), µ > 0. The parameter µ > 0 is used as the statistical model parameter 1 which is the expected value of each observation random variable Yi. (a) Form the maximum likelihood estimate μ^(y) in the model for the parameter μ and form the maximum likelihood estimator µ^(Y) in the model for the parameter μ. (b) Assume that n = intervals2 (min)) is 6 and the observed data (i.e., six dog encounter y=(y1,y2,...,y5,y6)=(2.25, 8.40, 5.10, 11.85, 3.30, 8.10). Use the data to calculate the ML estimate for the expected value μ of the intervals. Also consider what can be deduced from this: On the basis on this analysis of problem, do you think it is possible, i.e. likely, that expected value would actually be 3 (minutes). There is no "right answer" to this.
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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Transcribed Image Text:Person P is walking with his dog every day. He decides to model the time
intervals between dog encounters (i.e., how long does it take between two
encounters) with the independent random variables Y₁, . . ., Yn . Number of
dogs encounters per day, the number of customers entering a store per hour,
the number of bacteria per liter of water, etc. are classically modelled using
the Poisson distribution. With an exponential distribution, on the other hand, is
used for modelling intervals of between events and how long a light bulb lasts,
for example. After learning this, person P decides to model the observations
in such a way that the corresponding random variables Y1,..., Yn for a random
sample, that is, independent observation, of the exponential distribution
Exp(1/µ), µ > 0. The parameter µ > 0 is used as the statistical model
parameter 1 which is the expected value of each observation random variable
Yi.
(a) Form the maximum likelihood estimate μ^(y) in the model for the
parameter μ and form the maximum likelihood estimator µ^(Y) in the model for
the parameter μ.
(b) Assume that n
=
intervals2 (min)) is
6 and the observed data (i.e., six dog encounter
y=(y1,y2,...,y5,y6)=(2.25, 8.40, 5.10, 11.85, 3.30, 8.10).
Use the data to calculate the ML estimate for the expected value μ of the
intervals. Also consider what can be deduced from this: On the basis on this
analysis of problem, do you think it is possible, i.e. likely, that expected value
would actually be 3 (minutes). There is no "right answer" to this.
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