3. Consider the Poisson distribution with parameter μ, where > 0. Its probability mass function is Pμ(y) = e ¹μ/y! for y = 0, 1, 2,.... (a) Show that the Poisson distribution belongs to the exponential family of distribution with natural parameter n = log(u), where log(x) = ln(x), and show that its cumulant function is c(n) = en. (b) What is the carrier measure of the distribution? = fly (c) Use the cumulant function to show that the mean and the variance for this distribution are E[Y] = and V[Y] = μ, respectively. (d) Suppose that we have a random sample of size n = 75 from a Poisson distribution with mean . The sample mean of the 75 observations is y = 17.5. Use the score test to test Hou 20 against Hau 20. Give the observed value of the score test statistic, the p-value, and the conclusion at a = 5%.
3. Consider the Poisson distribution with parameter μ, where > 0. Its probability mass function is Pμ(y) = e ¹μ/y! for y = 0, 1, 2,.... (a) Show that the Poisson distribution belongs to the exponential family of distribution with natural parameter n = log(u), where log(x) = ln(x), and show that its cumulant function is c(n) = en. (b) What is the carrier measure of the distribution? = fly (c) Use the cumulant function to show that the mean and the variance for this distribution are E[Y] = and V[Y] = μ, respectively. (d) Suppose that we have a random sample of size n = 75 from a Poisson distribution with mean . The sample mean of the 75 observations is y = 17.5. Use the score test to test Hou 20 against Hau 20. Give the observed value of the score test statistic, the p-value, and the conclusion at a = 5%.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
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![3. Consider the Poisson distribution with parameter u, where u > 0. Its probability mass function is
P„(y) = e-"µ³/y!
for y = 0, 1, 2, ....
(a) Show that the Poisson distribution belongs to the exponential family of distribution with natural
parameter 7 = log(µ), where log(x) = ln(x), and show that its cumulant function is c(n) = e".
(b) What is the carrier measure of the distribution?
(c) Use the cumulant function to show that the mean and the variance for this distribution are E[Y] = µ,
and V[Y] = µ, respectively.
(d) Suppose that we have a random sample of size n = 75 from a Poisson distribution with mean u. The
sample mean of the 75 observations is j = 17.5. Use the score test to test
Ho : µ = 20 against
Ha : u + 20.
а
Give the observed value of the score test statistic, the p-value, and the conclusion at a =
5%.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f370799-6fa0-4692-86b4-b2c7db299570%2F7b4de22d-f796-43bf-a26c-d33b054d457e%2F4klis9_processed.png&w=3840&q=75)
Transcribed Image Text:3. Consider the Poisson distribution with parameter u, where u > 0. Its probability mass function is
P„(y) = e-"µ³/y!
for y = 0, 1, 2, ....
(a) Show that the Poisson distribution belongs to the exponential family of distribution with natural
parameter 7 = log(µ), where log(x) = ln(x), and show that its cumulant function is c(n) = e".
(b) What is the carrier measure of the distribution?
(c) Use the cumulant function to show that the mean and the variance for this distribution are E[Y] = µ,
and V[Y] = µ, respectively.
(d) Suppose that we have a random sample of size n = 75 from a Poisson distribution with mean u. The
sample mean of the 75 observations is j = 17.5. Use the score test to test
Ho : µ = 20 against
Ha : u + 20.
а
Give the observed value of the score test statistic, the p-value, and the conclusion at a =
5%.
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