Assume that you have random variable X with pdf or pmf f(x; θ1, . . ....
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Assume that you have random variable X with pdf or pmf f(x; θ1, . .
....
4. Let Xi,... . Xn be lid discrete uniform random variables with common pmf θ, with...
4. Let Xi,... . Xn be lid discrete uniform random variables with common pmf θ, with th θ) being {1, 2, . . .). Let T-max(X1, . .. , X e parameter space for (a) Derive the distribution of T. (Hint: use the edf approach). (b) Give the conditional distribution of Xi,... ,Xn given T-
, xn be a sample with joint pdf (or pmf) f(Xn10), θ 3. Let Xi, Θ C R. Suppose that {f(x,10) : θ E Θ} has monotone likelihood ratio (MLR) in T(Xn). Consider test function if T(%) > c Xn if T(%) <...
, xn be a sample with joint pdf (or pmf) f(Xn10), θ 3. Let Xi, Θ C R. Suppose that {f(x,10) : θ E Θ} has monotone likelihood ratio (MLR) in T(Xn). Consider test function if T(%) > c Xn if T(%) < c, where γ E [0, 1) and c 〉 0 are constants. Prove that the power function of φ(Xn) is non-decreasing in θ
, xn be a sample with joint pdf (or pmf) f(Xn10), θ 3. Let...
Let X be a random variable with p.d.f. f(x) = θx^(θ−1) , for 0 < x...
Let X be a random variable with p.d.f. f(x) = θx^(θ−1) , for 0 < x < 1. Let X1, ..., Xn denote a random sample of size n from this distribution. (a) Find E(X) [2] (b) Find the method of moment estimator of θ [2] (c) Find the maximum likelihood estimator of θ [3] (d) Use the following set of observations to obtain estimates of the method of moment and maximum likelihood estimators of θ. [1 each] 0.0256, 0.3051,...
Let X1,…, Xn be a sample of iid random variable with pdf f (x; ?) =...
Let X1,…, Xn be a sample of iid random variable with pdf f (x; ?) = 1/(2x−?+1) on S = {?, ? + 1, ? + 2,…} with Θ = ℕ. Determine a) a sufficient statistic for ?. b) F(1)(x). c) f(1)(x). d) E[X(1)].
Let X1,..., Xn be a random sample from the pdf f(x:0)-82-2, 0 < θ x <...
Let X1,..., Xn be a random sample from the pdf f(x:0)-82-2, 0 < θ x < oo. (a) Find the method of moments estimator of θ. (b) Find the maxinum likelihood estimator of θ
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,...
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...
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...
Let X1, ..., Xn be a random sample from a population with pdf f(x 1/8,0 <...
Let X1, ..., Xn be a random sample from a population with pdf f(x 1/8,0 < x < θ, zero elsewhere. Let Yi < < Y, be the order statistics. Show that Y/Yn and Yn are independent random variables
4. Let Xi,... . Xn be lid discrete uniform random variables with common pmf θ, with th θ) being {1, 2, . . .). Let T-max(X1, . .. , X e parameter space for (a) Derive the distribution of T. (Hint: use the edf approach). (b) Give the conditional distribution of Xi,... ,Xn given T-
, xn be a sample with joint pdf (or pmf) f(Xn10), θ 3. Let Xi, Θ C R. Suppose that {f(x,10) : θ E Θ} has monotone likelihood ratio (MLR) in T(Xn). Consider test function if T(%) > c Xn if T(%) < c, where γ E [0, 1) and c 〉 0 are constants. Prove that the power function of φ(Xn) is non-decreasing in θ
, xn be a sample with joint pdf (or pmf) f(Xn10), θ 3. Let...
Let X1,..., Xn be a random sample from the pdf f(x:0)-82-2, 0 < θ x < oo. (a) Find the method of moments estimator of θ. (b) Find the maxinum likelihood estimator of θ
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...
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...
Let X1, ..., Xn be a random sample from a population with pdf f(x 1/8,0 < x < θ, zero elsewhere. Let Yi < < Y, be the order statistics. Show that Y/Yn and Yn are independent random variables
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