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5. (Chapter 9.4, 9.5, 20 pt.) Y1, ..., Y. is a normally distributed random sample with...
Y1, Y2, ... Yn are a random sample from the Gamma distribution with parameters α and β
Y1, Y2, ... Yn are a random sample from the Gamma distribution with parameters α and β (a) Suppose that α-4 is known and β is unknown. Find a complete sufficient statistic for β. Find the MVUE of β. (Hint: What is E(Y)?) (b) Suppose that β = 4 is known and a is unknown. Find a complete sufficient statistic for α.
Let Y1,K,Y n denote a random sample from a Poisson distribution with parameter λ . a....
Let Y1,K,Y n denote a random sample from a Poisson distribution with parameter λ . a. Find a sufficient statistics for λ. b. Find the minimum variance unbiased estimator(MVUE) of λ2 .
8) Let Yi, X, denote a random sample from a normal distribution with mean μ and...
8) Let Yi, X, denote a random sample from a normal distribution with mean μ and variance σ , with known μ and unknown σ' . You are given that Σ(X-μ)2 is sufficient for σ a) Find El Σ(X-μ). |. Show all steps. Use the fact that: Var(Y)-E(P)-(BY)' i-1 b) Find the MVUE of σ.
6. Consider a sample X,... X, of normally distributed random variables with mean y and variance...
6. Consider a sample X,... X, of normally distributed random variables with mean y and variance op. Let 5 be the sample variance and suppose that n = 16. What is the value of c for which p[x - SS (C2 - 1)] = 95 ? be the 7. Consider a sample X,...,X, of normally distributed random variables with variance o? = 30. Let S sample variance and suppose that n-61. What is the value of c for which P...
10. (8 marks) Suppose Y, Y is a random sample of independent and identically distributed random...
10. (8 marks) Suppose Y, Y is a random sample of independent and identically distributed random variables with density function given by else a) (5 marks) By conditioning (definition 9.3) show that Uis sufficient for 0 b) (3 marks) By factorization (theorem 9.4) show that U- is sufficient for 0 Definition 9.3: Sufficiency The statistic U-g(, , X,) is said to be sufficient for θ if the conditional distribution of Y, Y, given U, does not depend on e Theorem...
4. Let X1, X2, ...,Xn be a random sample from a normal distribution with mean 0...
4. Let X1, X2, ...,Xn be a random sample from a normal distribution with mean 0 and unknown variance o2. (a) Show that U = <!-, X} is a sufficient statistic for o?. [4] (c) Show that the MLE of o2 is Ô = 2-1 X?. [4] (c) Calculate the mean and variance of Ô from (b). Explain why ő is also the MVUE of o2. [6]
7. A random sample of 20 stock return is believed to be normally distributed with mean u and variance ơ2. The retur...
7. A random sample of 20 stock return is believed to be normally distributed with mean u and variance ơ2. The returns. X. are recorded as follows: 0.03 0.090 0.022 0.100 0.0120.000.0160.1310.0380.038 0.107 0.165 0.102 0.0060.047 0.010 0.0710.094 0.029 0.057 By setting α-0.10, test the hypothesis Ho: σ2 0.01 against the alternative, H1:02 < 0.01 Determine the 95% confidence intervals for by assuming that, a. b. Ơ2 0.0 1, and σ2 is not known. I. 11,
7. A random sample...
Suppose that Y1 , Y2 ,..., Yn denote a random sample of size n from a...
Suppose
that Y1 , Y2 ,..., Yn denote a random sample of size n from a
normal population with mean μ and variance 2 .
Problem # 2: Suppose that Y , Y,,...,Y, denote a random sample of size n from a normal population with mean u and variance o . Then it can be shown that (n-1)S2 p_has a chi-square distribution with (n-1) degrees of freedom. o2 a. Show that S2 is an unbiased estimator of o. b....
with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a) Suppose that α 4 is known and β is unknown. Find a complete sufficient statistic for β. Find the MVUE of β....
with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a) Suppose that α 4 is known and β is unknown. Find a complete sufficient statistic for β. Find the MVUE of β. (Hint: What is E(Y)?) (b) Suppose that β 4 is known and α is unknown. Find a complete sufficient statistic for a.
with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a)...
5. Suppose that X, X, ..., X, is a random sample from a distribution with the...
5. Suppose that X, X, ..., X, is a random sample from a distribution with the density function (@+1)x®, if 0 < x <1 1 0, otherwise (where @ > -1 is unknown). (a) Show that the moments estimator of e is à 28-1 1-X (b) (c) (where X denotes the sample mean, as usual). Show that is a consistent estimator of e. U = - h, In X, is a sufficient statistic for 8. Is a function of U?...
8) Let Yi, X, denote a random sample from a normal distribution with mean μ and variance σ , with known μ and unknown σ' . You are given that Σ(X-μ)2 is sufficient for σ a) Find El Σ(X-μ). |. Show all steps. Use the fact that: Var(Y)-E(P)-(BY)' i-1 b) Find the MVUE of σ.
6. Consider a sample X,... X, of normally distributed random variables with mean y and variance op. Let 5 be the sample variance and suppose that n = 16. What is the value of c for which p[x - SS (C2 - 1)] = 95 ? be the 7. Consider a sample X,...,X, of normally distributed random variables with variance o? = 30. Let S sample variance and suppose that n-61. What is the value of c for which P...
10. (8 marks) Suppose Y, Y is a random sample of independent and identically distributed random variables with density function given by else a) (5 marks) By conditioning (definition 9.3) show that Uis sufficient for 0 b) (3 marks) By factorization (theorem 9.4) show that U- is sufficient for 0 Definition 9.3: Sufficiency The statistic U-g(, , X,) is said to be sufficient for θ if the conditional distribution of Y, Y, given U, does not depend on e Theorem...
4. Let X1, X2, ...,Xn be a random sample from a normal distribution with mean 0 and unknown variance o2. (a) Show that U = <!-, X} is a sufficient statistic for o?. [4] (c) Show that the MLE of o2 is Ô = 2-1 X?. [4] (c) Calculate the mean and variance of Ô from (b). Explain why ő is also the MVUE of o2. [6]
7. A random sample of 20 stock return is believed to be normally distributed with mean u and variance ơ2. The returns. X. are recorded as follows: 0.03 0.090 0.022 0.100 0.0120.000.0160.1310.0380.038 0.107 0.165 0.102 0.0060.047 0.010 0.0710.094 0.029 0.057 By setting α-0.10, test the hypothesis Ho: σ2 0.01 against the alternative, H1:02 < 0.01 Determine the 95% confidence intervals for by assuming that, a. b. Ơ2 0.0 1, and σ2 is not known. I. 11,
7. A random sample...
Suppose
that Y1 , Y2 ,..., Yn denote a random sample of size n from a
normal population with mean μ and variance 2 .
Problem # 2: Suppose that Y , Y,,...,Y, denote a random sample of size n from a normal population with mean u and variance o . Then it can be shown that (n-1)S2 p_has a chi-square distribution with (n-1) degrees of freedom. o2 a. Show that S2 is an unbiased estimator of o. b....
with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a) Suppose that α 4 is known and β is unknown. Find a complete sufficient statistic for β. Find the MVUE of β. (Hint: What is E(Y)?) (b) Suppose that β 4 is known and α is unknown. Find a complete sufficient statistic for a.
with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a)...
5. Suppose that X, X, ..., X, is a random sample from a distribution with the density function (@+1)x®, if 0 < x <1 1 0, otherwise (where @ > -1 is unknown). (a) Show that the moments estimator of e is à 28-1 1-X (b) (c) (where X denotes the sample mean, as usual). Show that is a consistent estimator of e. U = - h, In X, is a sufficient statistic for 8. Is a function of U?...
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