2. The negative binomial distribution has probability mass function (pmf), ƒ (y; r,p) = (³ + r − 1 ) (1 − p)'p', y=0,1,2,..., r>0,p>0. f( Assume that r is known. У - (a) Show that the pmf of the negative binomial has exponential family form, exp{[y b(0)]/a(o) + c(y,o)}, stating the canonical parameter 0, dispersion parameter and the functions b(0), a(o) and c(y, o). (b) Find the mean and variance of the negative binomial distribution in terms of r and p by using the functions b(0) and a(Ø). Hence, explain how the negative binomial distribution can be used to handle overdispersion that is not captured by the Poisson distribution.
2. The negative binomial distribution has probability mass function (pmf), ƒ (y; r,p) = (³ + r − 1 ) (1 − p)'p', y=0,1,2,..., r>0,p>0. f( Assume that r is known. У - (a) Show that the pmf of the negative binomial has exponential family form, exp{[y b(0)]/a(o) + c(y,o)}, stating the canonical parameter 0, dispersion parameter and the functions b(0), a(o) and c(y, o). (b) Find the mean and variance of the negative binomial distribution in terms of r and p by using the functions b(0) and a(Ø). Hence, explain how the negative binomial distribution can be used to handle overdispersion that is not captured by the Poisson distribution.
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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![2. The negative binomial distribution has probability mass function (pmf),
ƒ (y; r,p) = (³ + r − 1 ) (1 − p)'p', y=0,1,2,..., r>0,p>0.
f(
Assume that r is known.
У
-
(a) Show that the pmf of the negative binomial has exponential family form,
exp{[y b(0)]/a(o) + c(y,o)}, stating the canonical parameter 0, dispersion
parameter and the functions b(0), a(o) and c(y, o).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F008f8cce-1e45-43a4-8b17-46721d7357f5%2Fa0de7bbd-37eb-440d-84ef-1f2d9479b711%2Fzqhd8ut_processed.png&w=3840&q=75)
Transcribed Image Text:2. The negative binomial distribution has probability mass function (pmf),
ƒ (y; r,p) = (³ + r − 1 ) (1 − p)'p', y=0,1,2,..., r>0,p>0.
f(
Assume that r is known.
У
-
(a) Show that the pmf of the negative binomial has exponential family form,
exp{[y b(0)]/a(o) + c(y,o)}, stating the canonical parameter 0, dispersion
parameter and the functions b(0), a(o) and c(y, o).
Transcribed Image Text:(b) Find the mean and variance of the negative binomial distribution in terms of
r and p by using the functions b(0) and a(Ø). Hence, explain how the negative
binomial distribution can be used to handle overdispersion that is not captured
by the Poisson distribution.
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