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USU RNSAS t8 ahswer the following: 1. Suppose that we want to sample from θ/x~N(0,1)using the Met...
1. Suppose we are going to sample n individuals and ask each sampled person whether they...
1. Suppose we are going to sample n individuals and ask each sampled person whether they support policy A or not. Let Yi Y0 otherwise 1 if person i in the sample supports the policy, and (a) Assume Y1, , Yn are, conditional on θ. 1.1.d. binary random variables with expec- tation θ. Write down the joint distribution Pr(Yi-yi, . . . ,Ý,-yn(9) in a compact form. Also write down the form of Pr(> Ý,-y|0) (b) For the moment, suppose...
Suppose we want to estimate a parameter θ of a certain distribution and we have the following independent point estimates N(0+0.1,0.01) N(0, 0.04) B2 ~ a) What are the mean square errors for these po...
Suppose we want to estimate a parameter θ of a certain distribution and we have the following independent point estimates N(0+0.1,0.01) N(0, 0.04) B2 ~ a) What are the mean square errors for these point estimates? (4pts) b) Find a point estimate with mean square error less than or equal to 0.01. (2pts) c) Only use ël and Ộ2, find the unbiased estimator with the smallest variance possible. What is that estimator? What is the smallest variance? (6pts)
Suppose we...
We have n independent observations from a geometric distribution with unknown parameter θ. Po(X,-k-θ(1-0)4-1 for k-1,...
We have n independent observations from a geometric distribution with unknown parameter θ. Po(X,-k-θ(1-0)4-1 for k-1, 2, 3, . . . We wish to test the null hypothesis θ-1/2 versus the alternative θ 7|/2. we can show that the MLE θ-1/2. Write out the appropriate LRT statistic as a function of the r, the mean of the observations
FR2 (4+4+4 12 points) (a) Let XI, X2, X10 be a randoin sample from N(μι,σ?) and Yi, Y2, 10 , Y 15 be a random sample from N (μ2, σ2), where all parameters are unknown. Sup- pose Σ 1 (Xi X 2 0 321 (Y-...
FR2 (4+4+4 12 points) (a) Let XI, X2, X10 be a randoin sample from N(μι,σ?) and Yi, Y2, 10 , Y 15 be a random sample from N (μ2, σ2), where all parameters are unknown. Sup- pose Σ 1 (Xi X 2 0 321 (Y-Y )2-100. obtain a 99% confidence interval for σ of having the form b, 0o) for some number b (No derivation needed). (b) 60 random points are selected from the unit interval (r:0 . We want...
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...
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...
4. (Part 1)Suppose a random sample of size n is drawn from Unif(0, θ). We wish...
4. (Part 1)Suppose a random sample of size n is drawn from Unif(0, θ). We wish to test H0: θ = 3 vs. H1: θ > 3 using the critical region Xmax > c. If the test has α = 0.05 and β = 0.12681 when θ = 4, find the values of c and n that make this happen. (Part2) Write a simulation that checks your answer from question 4.
1. Suppose we are going to sample n individuals and ask each sampled person whether they support policy A or not. Let Yi Y0 otherwise 1 if person i in the sample supports the policy, and (a) Assume Y1, , Yn are, conditional on θ. 1.1.d. binary random variables with expec- tation θ. Write down the joint distribution Pr(Yi-yi, . . . ,Ý,-yn(9) in a compact form. Also write down the form of Pr(> Ý,-y|0) (b) For the moment, suppose...
Suppose we want to estimate a parameter θ of a certain distribution and we have the following independent point estimates N(0+0.1,0.01) N(0, 0.04) B2 ~ a) What are the mean square errors for these point estimates? (4pts) b) Find a point estimate with mean square error less than or equal to 0.01. (2pts) c) Only use ël and Ộ2, find the unbiased estimator with the smallest variance possible. What is that estimator? What is the smallest variance? (6pts)
Suppose we...
We have n independent observations from a geometric distribution with unknown parameter θ. Po(X,-k-θ(1-0)4-1 for k-1, 2, 3, . . . We wish to test the null hypothesis θ-1/2 versus the alternative θ 7|/2. we can show that the MLE θ-1/2. Write out the appropriate LRT statistic as a function of the r, the mean of the observations
FR2 (4+4+4 12 points) (a) Let XI, X2, X10 be a randoin sample from N(μι,σ?) and Yi, Y2, 10 , Y 15 be a random sample from N (μ2, σ2), where all parameters are unknown. Sup- pose Σ 1 (Xi X 2 0 321 (Y-Y )2-100. obtain a 99% confidence interval for σ of having the form b, 0o) for some number b (No derivation needed). (b) 60 random points are selected from the unit interval (r:0 . We want...
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...
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...
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