Please provide the detailed steps and CORRECT answers to the following question: 7.1.5 Suppose that (x1,...,...
Please provide the detailed steps and CORRECT answers to the following question:
7.1.5 Suppose that (x1,..., xn) is a sample from a Uniform[0, θ] distribution with θ > 0 unknown. If the prior distribution of θ is Gamma(α, β) , then obtain the form of the posterior density of θ.
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Please provide the detailed steps and CORRECT answers to the
following question:
7.1.5 Suppose that (x1,...,...
Let X1, . . . , Xn be a random sample following Gamma(2, β) for some...
Let X1, . . . , Xn be a random sample following Gamma(2, β) for some unknown parameter β > 0. (i) Now let’s think like a Bayesian. Consider a prior distribution of β ∼ Gamma(a, b) for some a, b > 0. Derive the posterior distribution of β given (X1, . . . , Xn) = (x1,...,xn). (j) What is the posterior Bayes estimator of β assuming squared error loss?
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...
3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume...
3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume the prior distribution of Of λ is is an exponential with mean 1, i.e. the prior pdi g(A) e-λ, λ > 0 Note that the exponential distribution is a special gamma distribution; and a general gamma distribution with parameters α > 0 and β > 0 has the pd.f. h(A; α, β)-16(. otherwise Also the mean of a gamma random variable with the pd.f.h(Χα,...
Specifically, suppose that Xi, X2, .., Xn denote n payments, modeled as iid random variables with...
Specifically, suppose that Xi, X2, .., Xn denote n payments, modeled as iid random variables with common Weibull pdf 0, otherwise, where m > 0 is known and θ is unknown. In turn, suppose that θ ~ IG(α, β), that is, θ has an inverted gamma (prior) pdf 0, otherwise (a) Prove that the inverted gamma IG(α, β) prior is a conjugate prior for the Weibull family above. (b) Suppose that m-2, α-05, and β-2. Here are n-10 insurance payments...
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...
Number 4 turns out to be an inverse gamma function with parameters alpha= n and beta=...
Number 4 turns out to be an inverse gamma function with
parameters alpha= n and beta= the sum of x sub i
PLEASE ANSWER #5 NOT #4
4. Suppose that X1,X2, 10 pts. the p.d.f. is given by form a random sample from a distribution for which where the unknown parameter θ > 0. Suppose also that the improper prior of θ is m(0) Find the posterior distribution π(θ x). Hint: The inverse gamina distribution from question 6 in Homework...
Let the random sample X1, . . . , Xn be taken from the Binomial distribution...
Let the random sample X1, . . . , Xn be taken from the Binomial distribution with parameter θ, which is unknown and must be estimated. Let the prior distribution of θ be the beta distribution with known parameters α > 0 and β > 0. Find the Bayes risk and the Bayes estimator using squared error loss. estimator of θ.
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 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?...
3. Let Xi,... , Xio be a random sample of size 10 from a gamma distribution...
3. Let Xi,... , Xio be a random sample of size 10 from a gamma distribution with α--3 and β 1/e. The prior distribution of θ is a gamma distribution with α-10 and B-2. Recall that the gamma density is given by elsewhere, (a) Find the posterior distribution of θ (b) If we observe 17, use the mean of the posterior distribution to give a point estimate of θ.
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...
3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume the prior distribution of Of λ is is an exponential with mean 1, i.e. the prior pdi g(A) e-λ, λ > 0 Note that the exponential distribution is a special gamma distribution; and a general gamma distribution with parameters α > 0 and β > 0 has the pd.f. h(A; α, β)-16(. otherwise Also the mean of a gamma random variable with the pd.f.h(Χα,...
Specifically, suppose that Xi, X2, .., Xn denote n payments, modeled as iid random variables with common Weibull pdf 0, otherwise, where m > 0 is known and θ is unknown. In turn, suppose that θ ~ IG(α, β), that is, θ has an inverted gamma (prior) pdf 0, otherwise (a) Prove that the inverted gamma IG(α, β) prior is a conjugate prior for the Weibull family above. (b) Suppose that m-2, α-05, and β-2. Here are n-10 insurance payments...
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...
Number 4 turns out to be an inverse gamma function with
parameters alpha= n and beta= the sum of x sub i
PLEASE ANSWER #5 NOT #4
4. Suppose that X1,X2, 10 pts. the p.d.f. is given by form a random sample from a distribution for which where the unknown parameter θ > 0. Suppose also that the improper prior of θ is m(0) Find the posterior distribution π(θ x). Hint: The inverse gamina distribution from question 6 in Homework...
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 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?...
3. Let Xi,... , Xio be a random sample of size 10 from a gamma distribution with α--3 and β 1/e. The prior distribution of θ is a gamma distribution with α-10 and B-2. Recall that the gamma density is given by elsewhere, (a) Find the posterior distribution of θ (b) If we observe 17, use the mean of the posterior distribution to give a point estimate of θ.
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