Let X11 X12X1, and X21, X22X2nz be two independent random samples of size ni and n₂ from two normal populations N(₁, 2) and N(2, 2) respectively. (a) Derive the maximum likelihood estimators (mle's) of all the parameters in the first population (X₁). Using analogy, state the mle's of the parameters of the second population. (b) Find the pooled estimator of the common variance when it is assumed that of = o=0². Suggest an unbiased estimator of o². (c) Assuming both ny and n₂ are small, suggest a pivotal function that can be used to derive a (1-a) x 100% confidence interval for (₁-₂) when o² = 0₂ = 0².
Let X11 X12X1, and X21, X22X2nz be two independent random samples of size ni and n₂ from two normal populations N(₁, 2) and N(2, 2) respectively. (a) Derive the maximum likelihood estimators (mle's) of all the parameters in the first population (X₁). Using analogy, state the mle's of the parameters of the second population. (b) Find the pooled estimator of the common variance when it is assumed that of = o=0². Suggest an unbiased estimator of o². (c) Assuming both ny and n₂ are small, suggest a pivotal function that can be used to derive a (1-a) x 100% confidence interval for (₁-₂) when o² = 0₂ = 0².
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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![Let X₁1 X12X1, and X21, X22X22 be two independent random samples of
size ni and n₂ from two normal populations N(₁, 2) and N(2, 2) respectively.
(a) Derive the maximum likelihood estimators (mle's) of all the parameters in the first
population (X₁). Using analogy, state the mle's of the parameters of the second
population.
(b) Find the pooled estimator of the common variance when it is assumed that of =
=o². Suggest an unbiased estimator of o².
(c) Assuming both ny and n₂ are small, suggest a pivotal function that can be used to
derive a (1-a) x 100% confidence interval for (₁-₂) when of = 0 = 0².](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb6b4902-24f8-4b5a-8abf-ecd52fa0b83e%2F0e1dac13-3fb4-498d-800e-620fccaaa775%2Fyayjgq3_processed.png&w=3840&q=75)
Transcribed Image Text:Let X₁1 X12X1, and X21, X22X22 be two independent random samples of
size ni and n₂ from two normal populations N(₁, 2) and N(2, 2) respectively.
(a) Derive the maximum likelihood estimators (mle's) of all the parameters in the first
population (X₁). Using analogy, state the mle's of the parameters of the second
population.
(b) Find the pooled estimator of the common variance when it is assumed that of =
=o². Suggest an unbiased estimator of o².
(c) Assuming both ny and n₂ are small, suggest a pivotal function that can be used to
derive a (1-a) x 100% confidence interval for (₁-₂) when of = 0 = 0².
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VIEWStep 2: Derive the maximum likelihood estimators for the population mean(X1).
VIEWStep 3: Derive the maximum likelihood estimators for the population variance.
VIEWStep 4: Determine the pooled estimator of the common variance.
VIEWStep 5: Derive (1 - α)*100% confidence interval for the population mean difference.
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