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9. Let X₁, X2,..., Xn be the random sample from a normal distribution with mean and variance o². Let X = ²/1 X₁ and S² = ¹₁ Σ²-1(X; - X)² be the sample mean and sample variance, respectively. n-1 n (a). Find the maximum likelihood estimate (MLE) of (u,0²) (b). Find the method of moments estimators (MME) of (μ, o²) (c). Derive the E(X), Var(X), and the distribution of sample mean X.

9. Let X₁, X2,..., Xn be the random sample from a normal distribution with mean and variance o². Let X = ²/1 X₁ and S² = ¹₁ Σ²-1(X; - X)² be the sample mean and sample variance, respectively. n-1 n (a). Find the maximum likelihood estimate (MLE) of (u,0²) (b). Find the method of moments estimators (MME) of (μ, o²) (c). Derive the E(X), Var(X), and the distribution of sample mean X.

MATLAB: An Introduction with Applications
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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9. Let X₁, X2,..., Xn be the random sample from a normal distribution with mean and variance o². Let X = 1/1 X₁ and S² = ¹₁-1(X₁ - X)² be the sample mean and sample variance, respectively. (a). Find the maximum likelihood estimate (MLE) of (μ, 0²) (b). Find the method of moments estimators (MME) of (μ,0²) (c). Derive the E(X), Var(X), and the distribution of sample mean X. (d). Find the E(S2). Is S22 an unbiased estimator of o²? (e). Write the distribution of (n-1)² (f). State the distribution of (X). Explain it briefly. Sn/√n (g). Find the (1-a) 100% confidence interval for μ. (h). Find the approximate (1-a) 100% confidence interval for u.
Transcribed Image Text:9. Let X₁, X2,..., Xn be the random sample from a normal distribution with mean and variance o². Let X = 1/1 X₁ and S² = ¹₁-1(X₁ - X)² be the sample mean and sample variance, respectively. (a). Find the maximum likelihood estimate (MLE) of (μ, 0²) (b). Find the method of moments estimators (MME) of (μ,0²) (c). Derive the E(X), Var(X), and the distribution of sample mean X. (d). Find the E(S2). Is S22 an unbiased estimator of o²? (e). Write the distribution of (n-1)² (f). State the distribution of (X). Explain it briefly. Sn/√n (g). Find the (1-a) 100% confidence interval for μ. (h). Find the approximate (1-a) 100% confidence interval for u.
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