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Suppose we are given n=12 observations from the N(μ,1) distribution: 15.644, 16.437, 17.287, 14.448, 15.308, 15.169,...
2. Suppose that we have n independent observations x1,..., xn from a normal distribution with mean...
2. Suppose that we have n independent observations x1,..., xn from a normal distribution with mean μ and variance σ, and we want to test (a) Find the maximum likelihood estimator of μ when the null hypothesis is true. (b) Calculate the Likelihood Ratio Test Statistic 2 lo g max L(μ, σ log | max L( 1) (c) Explain as clearly as you can what happens to T when our estimate of σ2 is less than 1. (d) Show that...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (iii) Suppose that we have a prior μ ~ N(a, b-1) where b > 0, Show that the prior distribution π(A) verifies r(11) x exp (iv) Show that the posterior π(μ|y) verifies (v) which distribution is π(μ|y)?
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and...
We are looking to calculate the power of a one-sided test from n independent observations Xi...
We are looking to calculate the power of a one-sided test from n independent observations Xi from a N(μ, σ2) distribution with a null hypothesis of Ho : μ-μ0 and an alternative H, : μ 〉 μ0. Supposing that we know σ2, we can form a test statistic T= and reject the null hypothesis when T 〉 1.645. This test has level α 0.05. We want a formula for the power of this test against the alternative that μ-74-This power...
2. Suppose that we have n independent observations x1,..., xn from a normal distribution with mean...
2. Suppose that we have n independent observations x1,..., xn from a normal distribution with mean μ and variance σ, and we want to test (a) Find the maximum likelihood estimator of μ when the null hypothesis is true. (b) Calculate the Likelihood Ratio Test Statistic 2 lo g max L(μ, σ log | max L( 1) (c) Explain as clearly as you can what happens to T when our estimate of σ2 is less than 1. (d) Show that...
3. Suppose that Xi,.... Xn is a random sample from a uniform distribution over [0,0) That...
3. Suppose that Xi,.... Xn is a random sample from a uniform distribution over [0,0) That is, 0 elsewhere Also suppose that the prior distribution of θ is a Pareto distribution with density 0 elsewhere where θ0 > 0 and α > 1. (a) Determine (b) Show , θ > max(T1 , . . . ,Zn,%) and hence deduce the posterior density of θ given x, . . . ,Zn is (c) Compute the mean of the posterior distribution and...
6. Suppose we observe Y,... Yn from a normal distribution with unknown parameters such that Y...
6. Suppose we observe Y,... Yn from a normal distribution with unknown parameters such that Y 24, s2 36, and n 15. (a) Find the rejection region of a level α-0.05 test of H0 : μ-20 vs. H1 : μ * 20. Would this test reject with the given data? (b) Find the rejection region of a level α -0.05 test of Ho : μ < 20 vs. H1 : μ > 20 would this test reject with the given...
We are looking to calculate the power of a one-sided test from n independent observations from...
We are looking to calculate the power of a one-sided test from n independent observations from a N(μ, σ2) distribution with a null hypothesis of Ho : μ-μο and an alternative H1 : μ > μο. Supposing that we know σ2, we can form a test statistic o/Vn and reject the null hypothesis when T > 1.645. This test has level α = 0.05. We want a formula for the power of this test against the alternative that μ-A-This power...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (vi) Suppose that ( of y with a -ab1. Suppose that you observe a realization Compute the posterior distribution value of 1. π(μ|1) and explain how it relates to π(μ). vii) Suppose now that you observe a second realization of y with a value of -1. Update the posterior π(p11) to incorporate this new information.
Bayesian...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (vi) Suppose that ( of y with a -ab1. Suppose that you observe a realization Compute the posterior distribution value of 1. π(μ|1) and explain how it relates to π(μ). vii) Suppose now that you observe a second realization of y with a value of -1. Update the posterior π(p11) to incorporate this new information.
Bayesian...
6. We are given 5 observations, namely (-51.2, 42.88,-35.15, 41.04, -13.68] for the surface temperatures on...
6. We are given 5 observations, namely (-51.2, 42.88,-35.15, 41.04, -13.68] for the surface temperatures on an asteroid. We will assume that these are independent observations from a N(μ, σ*) distribution with given standard deviation σ-49. Suppose further that we have a prior distribution on μ that is Normal with m ean 0 and standard deviation 36. (a) Find the posterior distribution of μ, and the Bayes estimate of μ, the mean surface temperature. (b) Give a 96% Highest Posterior...
2. Suppose that we have n independent observations x1,..., xn from a normal distribution with mean μ and variance σ, and we want to test (a) Find the maximum likelihood estimator of μ when the null hypothesis is true. (b) Calculate the Likelihood Ratio Test Statistic 2 lo g max L(μ, σ log | max L( 1) (c) Explain as clearly as you can what happens to T when our estimate of σ2 is less than 1. (d) Show that...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (iii) Suppose that we have a prior μ ~ N(a, b-1) where b > 0, Show that the prior distribution π(A) verifies r(11) x exp (iv) Show that the posterior π(μ|y) verifies (v) which distribution is π(μ|y)?
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and...
We are looking to calculate the power of a one-sided test from n independent observations Xi from a N(μ, σ2) distribution with a null hypothesis of Ho : μ-μ0 and an alternative H, : μ 〉 μ0. Supposing that we know σ2, we can form a test statistic T= and reject the null hypothesis when T 〉 1.645. This test has level α 0.05. We want a formula for the power of this test against the alternative that μ-74-This power...
2. Suppose that we have n independent observations x1,..., xn from a normal distribution with mean μ and variance σ, and we want to test (a) Find the maximum likelihood estimator of μ when the null hypothesis is true. (b) Calculate the Likelihood Ratio Test Statistic 2 lo g max L(μ, σ log | max L( 1) (c) Explain as clearly as you can what happens to T when our estimate of σ2 is less than 1. (d) Show that...
3. Suppose that Xi,.... Xn is a random sample from a uniform distribution over [0,0) That is, 0 elsewhere Also suppose that the prior distribution of θ is a Pareto distribution with density 0 elsewhere where θ0 > 0 and α > 1. (a) Determine (b) Show , θ > max(T1 , . . . ,Zn,%) and hence deduce the posterior density of θ given x, . . . ,Zn is (c) Compute the mean of the posterior distribution and...
6. Suppose we observe Y,... Yn from a normal distribution with unknown parameters such that Y 24, s2 36, and n 15. (a) Find the rejection region of a level α-0.05 test of H0 : μ-20 vs. H1 : μ * 20. Would this test reject with the given data? (b) Find the rejection region of a level α -0.05 test of Ho : μ < 20 vs. H1 : μ > 20 would this test reject with the given...
We are looking to calculate the power of a one-sided test from n independent observations from a N(μ, σ2) distribution with a null hypothesis of Ho : μ-μο and an alternative H1 : μ > μο. Supposing that we know σ2, we can form a test statistic o/Vn and reject the null hypothesis when T > 1.645. This test has level α = 0.05. We want a formula for the power of this test against the alternative that μ-A-This power...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (vi) Suppose that ( of y with a -ab1. Suppose that you observe a realization Compute the posterior distribution value of 1. π(μ|1) and explain how it relates to π(μ). vii) Suppose now that you observe a second realization of y with a value of -1. Update the posterior π(p11) to incorporate this new information.
Bayesian...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (vi) Suppose that ( of y with a -ab1. Suppose that you observe a realization Compute the posterior distribution value of 1. π(μ|1) and explain how it relates to π(μ). vii) Suppose now that you observe a second realization of y with a value of -1. Update the posterior π(p11) to incorporate this new information.
Bayesian...
6. We are given 5 observations, namely (-51.2, 42.88,-35.15, 41.04, -13.68] for the surface temperatures on an asteroid. We will assume that these are independent observations from a N(μ, σ*) distribution with given standard deviation σ-49. Suppose further that we have a prior distribution on μ that is Normal with m ean 0 and standard deviation 36. (a) Find the posterior distribution of μ, and the Bayes estimate of μ, the mean surface temperature. (b) Give a 96% Highest Posterior...
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