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Consider the hierarchical Bayes model (a) Show that the conditional pdf g(ply, 0) is the pdf of a beta distribution with parameters (b) Show that the conditional pdf g(θ|y, p) is the pdf of a gamma d...
STAT 140 Suppose that X have a gamma distribution with parameters a = 2 and θ=...
STAT 140 Suppose that X have a gamma distribution with parameters a = 2 and θ= 3, and suppose that the conditional distribution of Y given X=x, is uniform between 0 and x. (1) Find E(Y) and Var(Y). (2) Find the Moment Generating Function (MGF) of Y. What is the distribution of Y?
Recall from the slides that the Gamma distribution can be reparameterized using the two parameters The pdf looks like t...
Recall from the slides that the Gamma distribution can be reparameterized using the two parameters The pdf looks like the shape parameter, and μ the mean. 02 Let 6-( (^ ) and rewrite this as the pdf of a 2-parameter exponential family. Enter η (0)-T (y) below. θ =
Recall from the slides that the Gamma distribution can be reparameterized using the two parameters The pdf looks like the shape parameter, and μ the mean. 02 Let 6-( (^ )...
5. Consider a random sample Y1, . . . , Yn from a distribution with pdf f(y|θ) = 1 θ 2 xe−x/θ , 0 < x < ∞. Calcula...
5. Consider a random sample Y1, . . . , Yn from a distribution with pdf f(y|θ) = 1 θ 2 xe−x/θ , 0 < x < ∞. Calculate the ML estimator of θ. 6. Consider the pdf g(y|α) = c(1 + αy2 ), −1 < y < 1. (a) Show that g(y|α) is a pdf when c = 3 6 + 2α . (b) Calculate E(Y ) and E(Y 2 ). Referencing your calculations, explain why M1 can’t be...
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...
Show that the sum of the observations of a random sample of size n from gamma distribution with parameters 1 and θ (so f(x:0)-e-",x > 0 ) is sufficient for θ, using the definition ofsuficiency...
Show that the sum of the observations of a random sample of size n from gamma distribution with parameters 1 and θ (so f(x:0)-e-",x > 0 ) is sufficient for θ, using the definition ofsuficiency. Then show that the mle of θ is a function of the sufficient x10 statistic.
Show that the sum of the observations of a random sample of size n from gamma distribution with parameters 1 and θ (so f(x:0)-e-",x > 0 ) is sufficient for...
Show that the sum of the observations of a random sample of size n from gamma distribution with parameters 1 and θ (so f(x:0)-e-re, x > 0 ) is sufficient for θ, using x/θ the definition ofsuficien...
Show that the sum of the observations of a random sample of size n from gamma distribution with parameters 1 and θ (so f(x:0)-e-re, x > 0 ) is sufficient for θ, using x/θ the definition ofsuficiency. Then show that the mle of θ is a function of the sufficient statistic.
Show that the sum of the observations of a random sample of size n from gamma distribution with parameters 1 and θ (so f(x:0)-e-re, x > 0 ) is...
Recall that if X has a beta(a, B) distribution, then the probability density function (pdf) of...
Recall that if X has a beta(a, B) distribution, then the probability density function (pdf) of X is where α > 0 and β > 0. In this problem, we are going to consider the beta subfamily where α-β θ. Let X1, X2, , Xn denote an iid sample from a beta(8,9) distribution. (b) The two-dimensional statistic nm 27 is also a sufficient statistic for θ. What must be true about the conditional distribution (c) Show that T* (X) is...
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each oth...
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each other. Given an outcome x-(xi, , X.), the objective is to estimate both λ and p. Consider the following hierarchical Bayesian model: P U(0, 1) alp) Gammala, b) rlp.i)~Ber(p independently (x,lr.λ.Ρ) ~ Poiar.) independently . where r () and a and b are known parameters. We...
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each oth...
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each other. Given an outcome x-(xi, , X.), the objective is to estimate both λ and p. Consider the following hierarchical Bayesian model: P U(0, 1) alp) Gammala, b) rlp.i)~Ber(p independently (x,lr.λ.Ρ) ~ Poiar.) independently . where r () and a and b are known parameters. We...
Suppose y has a「(1,1) distribution while X given y has the conditional pdf elsewhere 0 Note that both the pdf of Y and the conditional pdf are easy to simulate. (a) Set up the following algorithm to...
Suppose y has a「(1,1) distribution while X given y has the conditional pdf elsewhere 0 Note that both the pdf of Y and the conditional pdf are easy to simulate. (a) Set up the following algorithm to generate a stream of iid observations with pdf fx(x) 1. Generate y ~ fy(y). 2. Generate X~fxy(XY), (b) How would you estimate E[X]?
Suppose y has a「(1,1) distribution while X given y has the conditional pdf elsewhere 0 Note that both the pdf...
Recall from the slides that the Gamma distribution can be reparameterized using the two parameters The pdf looks like the shape parameter, and μ the mean. 02 Let 6-( (^ ) and rewrite this as the pdf of a 2-parameter exponential family. Enter η (0)-T (y) below. θ =
Recall from the slides that the Gamma distribution can be reparameterized using the two parameters The pdf looks like the shape parameter, and μ the mean. 02 Let 6-( (^ )...
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...
Show that the sum of the observations of a random sample of size n from gamma distribution with parameters 1 and θ (so f(x:0)-e-",x > 0 ) is sufficient for θ, using the definition ofsuficiency. Then show that the mle of θ is a function of the sufficient x10 statistic.
Show that the sum of the observations of a random sample of size n from gamma distribution with parameters 1 and θ (so f(x:0)-e-",x > 0 ) is sufficient for...
Show that the sum of the observations of a random sample of size n from gamma distribution with parameters 1 and θ (so f(x:0)-e-re, x > 0 ) is sufficient for θ, using x/θ the definition ofsuficiency. Then show that the mle of θ is a function of the sufficient statistic.
Show that the sum of the observations of a random sample of size n from gamma distribution with parameters 1 and θ (so f(x:0)-e-re, x > 0 ) is...
Recall that if X has a beta(a, B) distribution, then the probability density function (pdf) of X is where α > 0 and β > 0. In this problem, we are going to consider the beta subfamily where α-β θ. Let X1, X2, , Xn denote an iid sample from a beta(8,9) distribution. (b) The two-dimensional statistic nm 27 is also a sufficient statistic for θ. What must be true about the conditional distribution (c) Show that T* (X) is...
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each other. Given an outcome x-(xi, , X.), the objective is to estimate both λ and p. Consider the following hierarchical Bayesian model: P U(0, 1) alp) Gammala, b) rlp.i)~Ber(p independently (x,lr.λ.Ρ) ~ Poiar.) independently . where r () and a and b are known parameters. We...
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each other. Given an outcome x-(xi, , X.), the objective is to estimate both λ and p. Consider the following hierarchical Bayesian model: P U(0, 1) alp) Gammala, b) rlp.i)~Ber(p independently (x,lr.λ.Ρ) ~ Poiar.) independently . where r () and a and b are known parameters. We...
Suppose y has a「(1,1) distribution while X given y has the conditional pdf elsewhere 0 Note that both the pdf of Y and the conditional pdf are easy to simulate. (a) Set up the following algorithm to generate a stream of iid observations with pdf fx(x) 1. Generate y ~ fy(y). 2. Generate X~fxy(XY), (b) How would you estimate E[X]?
Suppose y has a「(1,1) distribution while X given y has the conditional pdf elsewhere 0 Note that both the pdf...
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