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(1:0) = 0e- I> 0. The four most recent intervals between eruptions (in minutes) are I₁ = 32, I₂ = 10, Iz = 28, I₁ = 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) = 8e-1300 (b) Show that L'(0) is of the form L'(0) = 0'e-100 (4- 1300). (c) Show that the maximum likelihood estimate of 0 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(1:0) = 0e- I> 0. The four most recent intervals between eruptions (in minutes) are I₁ = 32, I₂ = 10, Iz = 28, I₁ = 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) = 8e-1300 (b) Show that L'(0) is of the form L'(0) = 0'e-100 (4- 1300). (c) Show that the maximum likelihood estimate of 0 based on the data is @ 0.0308 making your argument clear.
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.1: Measures Of Center
Problem 9PPS
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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. The probability density function of this distribution is given by
f(x: 0) =
- ве-вх
I > 0.
The four most recent intervals between eruptions (in minutes) are
I₁ = 32, I₂ = 10, z³ = 28, IĄ = 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) = 0-1300
(b) Show that L'() is of the form
-1308
L'(0) = 0³e¹(4 – 1300).
(c) Show that the maximum likelihood estimate of 9 based on the
data is 0.0308 making your argument clear.
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