Consider the following events: A: X1, X2, distribution Xn are independent and come from a Rayleigh with pdf f₁(2) = xp (-2²) 0₁ 201 AC: X1, X2,..., Xn are independent and come from an exponential distribution with pdf f2(x) = 0₂ exp(-0₂x) on r > 0. (a) Show that the Bayes factor, B(x, A), for event A against event A is B(x, A) = on r > 0. exp(02 Σκι - 0 02 and 2 (b) Let the parameters, 0₁ and 02, for the Rayleigh and exponential distributions, be estimated by their MLES, ₁ and 2, respectively. It can be shown (and you do not need to do so) that these MLEs are 0₁ = n 201 2n = Show that when 0₁-1 and 02 = 2, the exponential term of the Bayes factor is 1 and thus vanishes from the Bayes factor. That is, show that exp (0₂ Σ 2₁ - Σ10, ²²) - 02 = 1.
Consider the following events: A: X1, X2, distribution Xn are independent and come from a Rayleigh with pdf f₁(2) = xp (-2²) 0₁ 201 AC: X1, X2,..., Xn are independent and come from an exponential distribution with pdf f2(x) = 0₂ exp(-0₂x) on r > 0. (a) Show that the Bayes factor, B(x, A), for event A against event A is B(x, A) = on r > 0. exp(02 Σκι - 0 02 and 2 (b) Let the parameters, 0₁ and 02, for the Rayleigh and exponential distributions, be estimated by their MLES, ₁ and 2, respectively. It can be shown (and you do not need to do so) that these MLEs are 0₁ = n 201 2n = Show that when 0₁-1 and 02 = 2, the exponential term of the Bayes factor is 1 and thus vanishes from the Bayes factor. That is, show that exp (0₂ Σ 2₁ - Σ10, ²²) - 02 = 1.
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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Question
Do part b in detail
Transcribed Image Text:Consider the following events:
A: X1, X2, ...,X, are independent and come from a Rayleigh
distribution with pdf
fi(x) =
- exp
on r> 0.
01
201
A° : X1, X2,.., Xn are independent and come from an exponential
distribution with pdf
f2(x) = 02 exp (-02x)
on r> 0.
(a) Show that the Bayes factor, B(x, A), for event A against event A is
B(x, A) =
exp ( 02 > ri
201
(b) Let the parameters, 61 and 02, for the Rayleigh and exponential
distributions, be estimated by their MLES, 61 and 02, respectively. It
can be shown (and you do not need to do so) that these MLES are
and 02
2n
Show that when 61 01 and 02 = 02, the exponential term of the Bayes
factor is 1 and thus vanishes from the Bayes factor. That is, show that
exp ( 02>ri
= 1.
201
i=1
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