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here WKT prior is said to be conjugate prior if family of
posterior distribution is same as prior distribution. 
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Specifically, suppose that Xi, X2, .., Xn denote n payments, modeled as iid random variables with...
Problem 3.1 Suppose that XI, X2,... Xn is a random sample of size n is to...
Problem 3.1 Suppose that XI, X2,... Xn is a random sample of size n is to be taken from a Bermoulli distribution for which the value of the parameter θ is unknown, and the prior distribution of θ is a Beta(α,β) distribution. Represent the mean of this prior distribution as μο=α/(α+p). The posterior distribution of θ is Beta =e+ ΣΧ, β.-β+n-ΣΧ.) a) Show that the mean of the posterior distribution is a weighted average of the form where yn and...
8. Let X,.. , Xn be a random sampl le from a uniform(O, 0) distributio n....
8. Let X,.. , Xn be a random sampl le from a uniform(O, 0) distributio n. (a) Write down the likelihood function of (b) Suppose the prior distribution of θ is given by the Pareto(α, β) distribution with pdf αβα θα+1 , for θ > β > 0, α > 0 Derive the posterior distribution of 0 and conclude that the Pareto family of distributions is a conjugate prior for the uniform distribution.
Suppose that Xi, X2, , xn is an iid sample from a U(0,0) distribution, where θ...
Suppose that Xi, X2, , xn is an iid sample from a U(0,0) distribution, where θ 0. În turn, the parameter 0 is best regarded as a random variable with a Pareto(a, b) distribution, that is, bab 0, otherwise, where a 〉 0 and b 〉 0 are known. (a) Turn the "Bayesian crank" to find the posterior distribution of θ. I would probably start by working with a sufficient statistic (b) Find the posterior mean and use this as...
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...
Suppose that Xi, X2,..., Xn are independent random variables (not iid) with densities x, (x^, where...
Suppose that Xi, X2,..., Xn are independent random variables (not iid) with densities x, (x^, where 6, > 0, for i-1, 2, , n. versus H1: not Ho (c) Suppose Ho is true so that the common distribution of X1, X2,..., Xn, now viewed as being conditional on 6, is described by where θ > 0. Identify a conjugate prior for 0. Specify any hyperparameters in your prior (pick values for fun if you want). Show how to carry out...
Suppose that Xi, X2,..., Xn is an iid sample from r > 0 where θ 0....
Suppose that Xi, X2,..., Xn is an iid sample from r > 0 where θ 0. Consider testing Ho : θ-Bo versus H1: θ (a) Derive a size α likelihood ratio test (LRT). (b) Derive the power function P(0) of the LRT. θο, where θο is known. (c) Now consider putting an inverse gamma prior distribution on θ, namely, 1 00), a 4a where a and b are known. Show how to carry out the Bayesian test (d) Is the...
6. L , Xn be a random sample from a population with pdf et X1,. . . 9x1, xe (0,1), 0, otherwise, where θ E Θ (0.00) (a) Find a confidence interval for θ with confidence coefficient 1-α by pivoting a...
6. L , Xn be a random sample from a population with pdf et X1,. . . 9x1, xe (0,1), 0, otherwise, where θ E Θ (0.00) (a) Find a confidence interval for θ with confidence coefficient 1-α by pivoting a random variable based on statistic T(X,)--Σ-1 log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval) (b) Find the shortest I-α confidence interval for θ of the form a/T, b/T, where T(X,)...
Suppose that X1, X2, ..., Xn are independent random variables (not iid) with densities ÍXi(z10,) -.2...
Suppose that X1, X2, ..., Xn are independent random variables (not iid) with densities ÍXi(z10,) -.2 e _ θ:/z1(z > 0), where θί 〉 0, for i = 1, 2, , n. (a) Derive the form of the likelihood ratio test (LRT) statistic for testing versuS H1: not Ho. You do not have to find the distribution of the likelihood ratio test (LRT) statistic under Ho- Just find the form of the statistic. (b) From your result in part (a),...
6. L , Xn be a random sample from a population with pdf et X1,. . . 9x1, xe (0,1), 0, otherwise, where θ E Θ (0.00) (a)...
6. L , Xn be a random sample from a population with pdf et X1,. . . 9x1, xe (0,1), 0, otherwise, where θ E Θ (0.00) (a) Find a confidence interval for θ with confidence coefficient 1-α by pivoting a random variable based on statistic T(X,)--Σ-1 log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval) (b) Find the shortest I-α confidence interval for θ of the form a/T, b/T, where T(X,)...
Suppose observations X1, X2,.. are recorded. We assume these to be conditionally independent and exponen- tially...
Suppose observations X1, X2,.. are recorded. We assume these to be conditionally independent and exponen- tially distributed given a parameter θ: Xi ~' Exponential(θ), for all i 1, . . . , n. The exponential distribution is controlled by one rate parameter θ > 0, and its density is for r ER+ 1. Plot the graph of p(x:0) for θ 1 in the interval x E [0,4] 2. What is the visual representation of the likelihood of individual data points?...
Problem 3.1 Suppose that XI, X2,... Xn is a random sample of size n is to be taken from a Bermoulli distribution for which the value of the parameter θ is unknown, and the prior distribution of θ is a Beta(α,β) distribution. Represent the mean of this prior distribution as μο=α/(α+p). The posterior distribution of θ is Beta =e+ ΣΧ, β.-β+n-ΣΧ.) a) Show that the mean of the posterior distribution is a weighted average of the form where yn and...
8. Let X,.. , Xn be a random sampl le from a uniform(O, 0) distributio n. (a) Write down the likelihood function of (b) Suppose the prior distribution of θ is given by the Pareto(α, β) distribution with pdf αβα θα+1 , for θ > β > 0, α > 0 Derive the posterior distribution of 0 and conclude that the Pareto family of distributions is a conjugate prior for the uniform distribution.
Suppose that Xi, X2, , xn is an iid sample from a U(0,0) distribution, where θ 0. În turn, the parameter 0 is best regarded as a random variable with a Pareto(a, b) distribution, that is, bab 0, otherwise, where a 〉 0 and b 〉 0 are known. (a) Turn the "Bayesian crank" to find the posterior distribution of θ. I would probably start by working with a sufficient statistic (b) Find the posterior mean and use this as...
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...
Suppose that Xi, X2,..., Xn are independent random variables (not iid) with densities x, (x^, where 6, > 0, for i-1, 2, , n. versus H1: not Ho (c) Suppose Ho is true so that the common distribution of X1, X2,..., Xn, now viewed as being conditional on 6, is described by where θ > 0. Identify a conjugate prior for 0. Specify any hyperparameters in your prior (pick values for fun if you want). Show how to carry out...
Suppose that Xi, X2,..., Xn is an iid sample from r > 0 where θ 0. Consider testing Ho : θ-Bo versus H1: θ (a) Derive a size α likelihood ratio test (LRT). (b) Derive the power function P(0) of the LRT. θο, where θο is known. (c) Now consider putting an inverse gamma prior distribution on θ, namely, 1 00), a 4a where a and b are known. Show how to carry out the Bayesian test (d) Is the...
6. L , Xn be a random sample from a population with pdf et X1,. . . 9x1, xe (0,1), 0, otherwise, where θ E Θ (0.00) (a) Find a confidence interval for θ with confidence coefficient 1-α by pivoting a random variable based on statistic T(X,)--Σ-1 log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval) (b) Find the shortest I-α confidence interval for θ of the form a/T, b/T, where T(X,)...
Suppose that X1, X2, ..., Xn are independent random variables (not iid) with densities ÍXi(z10,) -.2 e _ θ:/z1(z > 0), where θί 〉 0, for i = 1, 2, , n. (a) Derive the form of the likelihood ratio test (LRT) statistic for testing versuS H1: not Ho. You do not have to find the distribution of the likelihood ratio test (LRT) statistic under Ho- Just find the form of the statistic. (b) From your result in part (a),...
6. L , Xn be a random sample from a population with pdf et X1,. . . 9x1, xe (0,1), 0, otherwise, where θ E Θ (0.00) (a) Find a confidence interval for θ with confidence coefficient 1-α by pivoting a random variable based on statistic T(X,)--Σ-1 log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval) (b) Find the shortest I-α confidence interval for θ of the form a/T, b/T, where T(X,)...
Suppose observations X1, X2,.. are recorded. We assume these to be conditionally independent and exponen- tially distributed given a parameter θ: Xi ~' Exponential(θ), for all i 1, . . . , n. The exponential distribution is controlled by one rate parameter θ > 0, and its density is for r ER+ 1. Plot the graph of p(x:0) for θ 1 in the interval x E [0,4] 2. What is the visual representation of the likelihood of individual data points?...
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