Let Xi , i = 1, · · · , n be a random sample from...
Let Xi , i = 1, · · · , n be a random sample from Poisson(θ) with pdf f(x|θ) = e −θ θ x x! , x = 0, 1, 2, · · · .
(a) Find the posterior distribution for θ when the prior is an exponential distribution with mean 1;
(b) Find the Bayesian estimator under the square loss function.
(c) Find a 95% credible interval for the parameter θ for the sample x1 = 2, x2 = 4, x3 = 1, x4 = 3. Note that you may need to use a statistical tool to find the percentiles corresponding to the the posterior distribution
Homework Answers
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...
Please answer the question clearly. Consider a random sample of size n from a Poisson population...
Please answer the question clearly.
Consider a random sample of size n from a Poisson population with parameter λ (a) Find the method of moments estimator for λ. (b) Find the maximum likelihood estimator for λ. Suppose X has a Poisson distribution and the prior distribution for its parameter A is a gamma distribution with parameters and β. (a) Show that the posterior distribution of A given X-x is a gamma distribution with parameters a +r and (b) Find the...
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...
can someone explains to me in details. i provided the answers yet i dont understand it realization of a random sample X...
can someone explains to me in details. i provided the answers
yet i dont understand it
realization of a random sample X1, X2, X3 from an N(u,a2) distri 2. Suppose that xi,x2, x3 are a bution with o known. If we decide to use Bayesian inference to obtain a reliable estimate for the parameter , and use 1 h(H) e 2 the prior distribution of the parameter i. Write down an expression for the posterior distribution (on-1) as of You...
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,...
Let Xi,..,Xn be a random sample from a population with pdf 0, x 0 is unknown. (a) Show that the G...
Please answer the following question and show every step. Thank
you.
Let Xi,..,Xn be a random sample from a population with pdf 0, x<0, where θ > 0 is unknown. (a) Show that the Gamma(a, b) prior with pdf 0, θ < 0. is a conjugate prior for θ (a > 0 and b > 0 are known constants). (b) Find the Bayes estimator of θ under square error loss. (c) Find the Bayes estimator of (2π-10)1/2 under square error...
Let X1, X2, ... , Xn be a random sample of size n from the exponential...
Let X1, X2, ... , Xn be a random sample of size n from the exponential distribution whose pdf is f(x; θ) = (1/θ)e^(−x/θ) , 0 < x < ∞, 0 <θ< ∞. Find the MVUE for θ. Let X1, X2, ... , Xn be a random sample of size n from the exponential distribution whose pdf is f(x; θ) = θe^(−θx) , 0 < x < ∞, 0 <θ< ∞. Find the MVUE for θ.
Let Y4 be the largest order statistic of a sample of size n = 4 from...
Let Y4 be the largest order statistic of a sample of size n = 4 from a distribution with uniform pdf f(x; θ) = 1/θ, 0 < x < θ, zero elsewhere. If the prior pdf of the parameter g(θ) = 2/θ3, 1 < θ < ∞, zero elsewhere, find the Bayesian estimator δ(Y4) of θ, based upon the sufficient statistic Y4, using the loss function |δ(y4) − θ|.
Please let me know how to solve 7.6.5. 6.5. Let Xi, X2,. .. X, be a random sample from a Poisson distribution with p...
Please let me know how to solve 7.6.5.
6.5. Let Xi, X2,. .. X, be a random sample from a Poisson distribution with parameter θ > 0. (a) Find the MVUE of P(X < 1)-(1 +0)c". Hint: Let u(x)-1, where Y = Σ1Xi. 1, zero elsewhere, and find Elu(Xi)|Y = y, xỉ (b) Express the MVUE as a function of the mle of θ. (c) Determine the asymptotic distribution of the mle of θ (d) Obtain the mle of P(X...
xercise 7.5: Suppose Xi, X2, ..., Xn are a random sample from the u distribution U(9-2 ,0+ ), where θ e (-00, Exercise...
xercise 7.5: Suppose Xi, X2, ..., Xn are a random sample from the u distribution U(9-2 ,0+ ), where θ e (-00, Exercise 7.5: Suppose X1, X2, . .. , sufficient for θ. a) Show that the smallest and largest of Xi, ..., Xn are jointliy (b) If p@-constant, θ e (-00, oo), is the prior distribution of θ, find its posterior distribution
xercise 7.5: Suppose Xi, X2, ..., Xn are a random sample from the u distribution U(9-2 ,0+...
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...
Please answer the question clearly.
Consider a random sample of size n from a Poisson population with parameter λ (a) Find the method of moments estimator for λ. (b) Find the maximum likelihood estimator for λ. Suppose X has a Poisson distribution and the prior distribution for its parameter A is a gamma distribution with parameters and β. (a) Show that the posterior distribution of A given X-x is a gamma distribution with parameters a +r and (b) Find the...
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...
can someone explains to me in details. i provided the answers
yet i dont understand it
realization of a random sample X1, X2, X3 from an N(u,a2) distri 2. Suppose that xi,x2, x3 are a bution with o known. If we decide to use Bayesian inference to obtain a reliable estimate for the parameter , and use 1 h(H) e 2 the prior distribution of the parameter i. Write down an expression for the posterior distribution (on-1) as of You...
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,...
Please answer the following question and show every step. Thank
you.
Let Xi,..,Xn be a random sample from a population with pdf 0, x<0, where θ > 0 is unknown. (a) Show that the Gamma(a, b) prior with pdf 0, θ < 0. is a conjugate prior for θ (a > 0 and b > 0 are known constants). (b) Find the Bayes estimator of θ under square error loss. (c) Find the Bayes estimator of (2π-10)1/2 under square error...
Please let me know how to solve 7.6.5.
6.5. Let Xi, X2,. .. X, be a random sample from a Poisson distribution with parameter θ > 0. (a) Find the MVUE of P(X < 1)-(1 +0)c". Hint: Let u(x)-1, where Y = Σ1Xi. 1, zero elsewhere, and find Elu(Xi)|Y = y, xỉ (b) Express the MVUE as a function of the mle of θ. (c) Determine the asymptotic distribution of the mle of θ (d) Obtain the mle of P(X...
xercise 7.5: Suppose Xi, X2, ..., Xn are a random sample from the u distribution U(9-2 ,0+ ), where θ e (-00, Exercise 7.5: Suppose X1, X2, . .. , sufficient for θ. a) Show that the smallest and largest of Xi, ..., Xn are jointliy (b) If p@-constant, θ e (-00, oo), is the prior distribution of θ, find its posterior distribution
xercise 7.5: Suppose Xi, X2, ..., Xn are a random sample from the u distribution U(9-2 ,0+...
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