Suppose that the interval between eruptions of a particular geyser can be modelled by an exponential distribution with an unknown parameter ✪ > 0. The probability density function of this distribution is given by f(x; 0) = 0e0¹, x > 0. The four most recent intervals between eruptions (in minutes) are ₁ = 32, ₂ = 10, 23=28, ₁ = 60; their values are to be treated as a random sample from the exponential distribution. (a) Show that the likelihood of 0 based on these data is given by L(0) = 0¹e-1300 (b) Show that L'(0) is of the form L'(0)=0³e-1300 (4-1300). (c) Show that the maximum likelihood estimate of based on the data is 0.0308 making your argument clear.
Suppose that the interval between eruptions of a particular geyser can be modelled by an exponential distribution with an unknown parameter ✪ > 0. The probability density function of this distribution is given by f(x; 0) = 0e0¹, x > 0. The four most recent intervals between eruptions (in minutes) are ₁ = 32, ₂ = 10, 23=28, ₁ = 60; their values are to be treated as a random sample from the exponential distribution. (a) Show that the likelihood of 0 based on these data is given by L(0) = 0¹e-1300 (b) Show that L'(0) is of the form L'(0)=0³e-1300 (4-1300). (c) Show that the maximum likelihood estimate of based on the data is 0.0308 making your argument clear.
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:Suppose that the interval between eruptions of a particular geyser can
be modelled by an exponential distribution with an unknown parameter
0 > 0. The probability density function of this distribution is given by
f(x; 0) = 0e 0¹, x > 0.
The four most recent intervals between eruptions (in minutes) are
x₁ = 32, x₂ = 10, x3 = 28, x4 = 60;
their values are to be treated as a random sample from the exponential
distribution.
(a) Show that the likelihood of based on these data is given by
L(0) 04-1306
=
(b) Show that L'(0) is of the form
L'(0) = 0³ e 1300 (4- 1300).
(c) Show that the maximum likelihood estimate of 0 based on the
data is ~ 0.0308 making your argument clear.
(d) Explain in detail how the maximum likelihood estimate of that
you have just obtained in part (c) relates to the maximum
likelihood estimator of for an exponential distribution.
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