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|.
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Let Y4 be the largest order statistic of a sample of size n = 4
from...
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...
Let X,X,, X, be a random sample of size 3 from a uniform distribution having pdf /(x:0) = θ,0 < x < 0,0 < θ, and let):く,), be the corresponding order statistics. a. Show that 2Y, is an unbi...
Let X,X,, X, be a random sample of size 3 from a uniform distribution having pdf /(x:0) = θ,0 < x < 0,0 < θ, and let):く,), be the corresponding order statistics. a. Show that 2Y, is an unbiased estimator of 0 and find its variance. b. Y is a sufficient statistic for 8. Determine the mean and variance of Y c. Determine the joint pdf of Y, and Y,, and use it to find the conditional expectation Find the...
15. (30 points) Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample of size n = 4 fr...
15. (30 points) Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample of size n = 4 from a distribution with p.d.f.f(x) 2x, 0 < x < 1, zero elsewhere. Evaluate E[Yalyj]. [Hint: First find the joint p.d.f. of Y3 and Y4, and then find the conditional p.d.f. of Y4 given Y3 y3]
15. (30 points) Let Y1
Let Y,,Y.,Y be a random sample of size n from a distribution having pdf a) Show that θ-Ymin is su...
Let Y,,Y.,Y be a random sample of size n from a distribution having pdf a) Show that θ-Ymin is sufficient for the threshold parameter θ. b) Show that Ymax is not sufficient for the threshold parameter θ.
Let Y,,Y.,Y be a random sample of size n from a distribution having pdf a) Show that θ-Ymin is sufficient for the threshold parameter θ. b) Show that Ymax is not sufficient for the threshold parameter θ.
Letter f and g only. 44 Let X,..., X. be a random sample from (a) Find a sufficient statistic. (b) Find a maximum-likel...
Letter f and g only.
44 Let X,..., X. be a random sample from (a) Find a sufficient statistic. (b) Find a maximum-likelihood estimator of θ. (c) Find a method-of-moments estimator of θ. (d) Is there a complete sufficient statistic? If so, find it. (e) Find the UMVUE of 0 if one exists. (f) Find the Pitman estimator for the location parameter θ. (g) Using the prior density g(0)--e-n,๑)(8), find the posterior Bayes estimator Of θ.
44 Let X,..., X....
6. Let X1, X2,.. , Xn denote a random sample of size n> 1 from a...
6. Let X1, X2,.. , Xn denote a random sample of size n> 1 from a distribution with pdf f(x; 6) = 6e-8, 0<x< 20, zero elsewhere, and 0 > 0. Le Y = x. (a) Show that Y is a sufficient and complete statistics for . (b) Prove that (n-1)/Y is an unbiased estimator of 0.
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random var...
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
. A random sample of size n is taken from a population that has a distri- bution with density fun...
. A random sample of size n is taken from a population that has a distri- bution with density function given by 0, elsewhere Find the likelihood function L(n v.. V ) -Using the factorization criterion, find a sufficient statistic for θ. Give your functions g(u, 0) and h(i, v2.. . n) - Use the fact that the mean of a random variable with distribution function above is to find the method of moment's estimator for θ. Explain how you...
Show that the mean X bar of a random sample of size n from a distribution...
Show that the mean X bar of a random sample of size n from a distribution having probability density function f(x;θ)=(1/θ)e-(x/θ) , ,0 < x < ∞ , 0 < θ < ∞ , zero elsewhere, is an unbiased estimator of θ and has variance θ2/n.
Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the distribution...
Let Y1<Y2<...<Yn be the
order statistics of a random sample of size n from the distribution
having p.d.f f(x) = e-y , 0<y<, zero elsewhere. Answer the following
questions.
(a) decide whether Z1 = Y2
and Z2=Y4-Y2 are
stochastically independent or not. (hint. first find the joint
p.d.f. of Y2 and Y4)
(b) show that
Z1 = nY1, Z2=
(n-1)(Y2-Y1),
Z3=(n-2)(Y3-Y2), ....,
Zn=Yn-Yn-1
are stocahstically
independent and that each Zi has the exponential
distribution.(hint use change of variable technique)
Let X,X,, X, be a random sample of size 3 from a uniform distribution having pdf /(x:0) = θ,0 < x < 0,0 < θ, and let):く,), be the corresponding order statistics. a. Show that 2Y, is an unbiased estimator of 0 and find its variance. b. Y is a sufficient statistic for 8. Determine the mean and variance of Y c. Determine the joint pdf of Y, and Y,, and use it to find the conditional expectation Find the...
15. (30 points) Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample of size n = 4 from a distribution with p.d.f.f(x) 2x, 0 < x < 1, zero elsewhere. Evaluate E[Yalyj]. [Hint: First find the joint p.d.f. of Y3 and Y4, and then find the conditional p.d.f. of Y4 given Y3 y3]
15. (30 points) Let Y1
Let Y,,Y.,Y be a random sample of size n from a distribution having pdf a) Show that θ-Ymin is sufficient for the threshold parameter θ. b) Show that Ymax is not sufficient for the threshold parameter θ.
Let Y,,Y.,Y be a random sample of size n from a distribution having pdf a) Show that θ-Ymin is sufficient for the threshold parameter θ. b) Show that Ymax is not sufficient for the threshold parameter θ.
Letter f and g only.
44 Let X,..., X. be a random sample from (a) Find a sufficient statistic. (b) Find a maximum-likelihood estimator of θ. (c) Find a method-of-moments estimator of θ. (d) Is there a complete sufficient statistic? If so, find it. (e) Find the UMVUE of 0 if one exists. (f) Find the Pitman estimator for the location parameter θ. (g) Using the prior density g(0)--e-n,๑)(8), find the posterior Bayes estimator Of θ.
44 Let X,..., X....
6. Let X1, X2,.. , Xn denote a random sample of size n> 1 from a distribution with pdf f(x; 6) = 6e-8, 0<x< 20, zero elsewhere, and 0 > 0. Le Y = x. (a) Show that Y is a sufficient and complete statistics for . (b) Prove that (n-1)/Y is an unbiased estimator of 0.
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
. A random sample of size n is taken from a population that has a distri- bution with density function given by 0, elsewhere Find the likelihood function L(n v.. V ) -Using the factorization criterion, find a sufficient statistic for θ. Give your functions g(u, 0) and h(i, v2.. . n) - Use the fact that the mean of a random variable with distribution function above is to find the method of moment's estimator for θ. Explain how you...
Let Y1<Y2<...<Yn be the
order statistics of a random sample of size n from the distribution
having p.d.f f(x) = e-y , 0<y<, zero elsewhere. Answer the following
questions.
(a) decide whether Z1 = Y2
and Z2=Y4-Y2 are
stochastically independent or not. (hint. first find the joint
p.d.f. of Y2 and Y4)
(b) show that
Z1 = nY1, Z2=
(n-1)(Y2-Y1),
Z3=(n-2)(Y3-Y2), ....,
Zn=Yn-Yn-1
are stocahstically
independent and that each Zi has the exponential
distribution.(hint use change of variable technique)
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