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A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS θ for the purpose of testing Ho : θ-2 against H, : θ < 2 at α : 0.10 level of significance. Suppose the decision rule is to be...
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.
Suppose that a random sample of size 36, Y1,Y2,...,Y36, is drawn from a uniform pdf de...
Suppose that a random sample of size 36, Y1,Y2,...,Y36, is drawn from a uniform pdf de ned over the interval (0, θ), where θ is unknown. Set up a large- sample sign test for deciding whether or not the 25th per- centile of the Y -distribution is equal to 6. Let α = 0.05. With what probability will your procedure commit a Type II error if 7 is the true 25th percentile?
8. Consider a random sample of size n from a distribution with pdf f(x) = 0...
8. Consider a random sample of size n from a distribution with pdf f(x) = 0 else (a) Find the pdf of the smallest order statistic, X(i) b) Find E() and Var(X)) c) Find the pdf of the largest order statistic, X(n)
2. (7 pts) Given the pdf f(x,0)- statistic Ymaz to test Ho : θ, θ > 0, .Take a sample of size 3 f...
2. (7 pts) Given the pdf f(x,0)- statistic Ymaz to test Ho : θ, θ > 0, .Take a sample of size 3 from this pdf. Use the 4,0 y 5 versus HA : θ > 5. (a) What is the decision rule when a 0.05. (b) Suppose θ-7, what is the Type II error for the test in part (a).
2. (7 pts) Given the pdf f(x,0)- statistic Ymaz to test Ho : θ, θ > 0, .Take a...
2. Let X1,.n be a random sample from the density 0 otherwise Suppose n = 2m+...
2. Let X1,.n be a random sample from the density 0 otherwise Suppose n = 2m+ 1 for some integer m. Let Y be the sample median and Z = max(Xi) be the sample maximum (a) Apply the usual formula for the density of an order statistic to show the density of Y is (b) Note that a beta random variable X has density f(x) = TaT(可 with mean μ = α/(a + β) and variance σ2 = αβ/((a +s+...
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...
4. (6 marks) Consider a random sample of size n from a distribution with pdf f(x:0)...
4. (6 marks) Consider a random sample of size n from a distribution with pdf f(x:0) 26-1 if 0 1 and zero otherwise; θ 0, Find the UMVUE of 1/θ x
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...
Suppose X, X2,... ,Xn represent a random sample from the pdf f(x;0) 2/02, 0 < x...
Suppose X, X2,... ,Xn represent a random sample from the pdf f(x;0) 2/02, 0 < x 〈 θ. Note, this pdf does not follow the regularity conditions. Nevertheless, find the MLE θη f . Make sure ElPn or a sample siZe n
MA2500/18 8. Let X be a random variable and let 'f(r; θ) be its PDF where...
MA2500/18 8. Let X be a random variable and let 'f(r; θ) be its PDF where θ is an unknown scalar parameter. We wish to test the simple null hypothesis Ho: 0 against the simple alternative Hi : θ-64. (a) Define the simple likelihood ratio test (SLRT) of Ho against H (b) Show that the SLRT is a most powerful test of Ho against H. (c) Let Xi, X2.... , X be a random sample of observations from the Poisson(e)...
8. Consider a random sample of size n from a distribution with pdf f(x) = 0 else (a) Find the pdf of the smallest order statistic, X(i) b) Find E() and Var(X)) c) Find the pdf of the largest order statistic, X(n)
2. (7 pts) Given the pdf f(x,0)- statistic Ymaz to test Ho : θ, θ > 0, .Take a sample of size 3 from this pdf. Use the 4,0 y 5 versus HA : θ > 5. (a) What is the decision rule when a 0.05. (b) Suppose θ-7, what is the Type II error for the test in part (a).
2. (7 pts) Given the pdf f(x,0)- statistic Ymaz to test Ho : θ, θ > 0, .Take a...
2. Let X1,.n be a random sample from the density 0 otherwise Suppose n = 2m+ 1 for some integer m. Let Y be the sample median and Z = max(Xi) be the sample maximum (a) Apply the usual formula for the density of an order statistic to show the density of Y is (b) Note that a beta random variable X has density f(x) = TaT(可 with mean μ = α/(a + β) and variance σ2 = αβ/((a +s+...
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...
4. (6 marks) Consider a random sample of size n from a distribution with pdf f(x:0) 26-1 if 0 1 and zero otherwise; θ 0, Find the UMVUE of 1/θ x
Suppose X, X2,... ,Xn represent a random sample from the pdf f(x;0) 2/02, 0 < x 〈 θ. Note, this pdf does not follow the regularity conditions. Nevertheless, find the MLE θη f . Make sure ElPn or a sample siZe n
MA2500/18 8. Let X be a random variable and let 'f(r; θ) be its PDF where θ is an unknown scalar parameter. We wish to test the simple null hypothesis Ho: 0 against the simple alternative Hi : θ-64. (a) Define the simple likelihood ratio test (SLRT) of Ho against H (b) Show that the SLRT is a most powerful test of Ho against H. (c) Let Xi, X2.... , X be a random sample of observations from the Poisson(e)...
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