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3. Again, let XXn be iid observations from the Uniform(0,0) distribution. (a) Find the joint pdf...
Again, let X1,..., Xn be iid observations from the Uniform(0,0) distribution. (a) Find the joint pdf...
Again, let X1,..., Xn be iid observations from the Uniform(0,0) distribution. (a) Find the joint pdf of Xi) and X(n)- (b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (c) It turns out, if Xi, . . . , xn (iid) Uniform(0,0), E(R)-θ . What happens to E® as n increases? Briefly explain in words why this makes sense intuitively.
Ques 3 (d) Suppose that n-10, and Xi Xio represent the waiting times that the 10...
Ques 3
(d) Suppose that n-10, and Xi Xio represent the waiting times that the 10 people must wait at a bus stop for their bus to arrive. Interpret the result of (c) in the context of this scenario be iid observations from the Uniform(0,0) distribution. 3. Again, let X..., X (a) Find the joint pdf of Xu) and X() (b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (e) It turns out, if X...
Only ques 4 (b) Define R = X(n)-X(1) as the sample range. Find the pdf of...
Only ques 4
(b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (c) It turns out, if Xi, . . . , Xn ~ (iid) Uniform(0,0), E(R)-θ . What happens to E(R) as n increases? Briefly explain in words why this makes sense intuitively. 4. Let X. Xn be a random sample from a population with pdf xotherwise Let Xa)<..< X(n) be the order statistics. Show that Xa)/X() and X(n) are independent random variables 5....
Problem 8: 5 points] Let Xi,.,.Xn be IID from a Uniform distribution on (-0,0) where 0...
Problem 8: 5 points] Let Xi,.,.Xn be IID from a Uniform distribution on (-0,0) where 0 0 is an unknown parameter (a) Find a minimal sufficient statistic T. (b) Define Show that T and V are independent.
PROBLEM 3 Let X1, X2,L , X, be iid observations from a distribution with pdf given...
PROBLEM 3 Let X1, X2,L , X, be iid observations from a distribution with pdf given by f(xl0)=0x0-, 0<x<1, 0<O<00. a) Find the maximum likelihood estimator of O. b) Find the moment estimator of 0. c) (Extra credit) Compare the mean squared error of the two estimators in (a) and (b). Which one is better? (5 points)
Suppose X1, X2, ..., Xn are independent and identically distributed (iid) with a Uniform -0,0 distri-...
Suppose X1, X2, ..., Xn are independent and identically distributed (iid) with a Uniform -0,0 distri- bution for some unknown e > 0, i.e., the Xi's have pdf Suppose X1, X2,..., Xn are independent and identically distributed (iid f(3) = S 20, if –0 < x < 0; 20 0, otherwise. (a) (4 pts) Briefly explain why or why not this is an exponential family (b) (5 pts) Find one meaningful sufficient statistic for 0. (By "meaningful”, I mean it...
1. Let X be an iid sample of size n from a continuous distribution with mean...
1. Let X be an iid sample of size n from a continuous distribution with mean /i, variance a2 and such that Xi e [0, 1] for all i e {1,...,n}. Let X = average. For a E (0,1), we wish to obtain a number q > 0 such that: (1/n) Xi be the sample Р(X € |и — 9. и + q) predict with probability approximately In other words, we wish to sample of size n, the average X...
Let Xi, , Xn be a sample from U(0,0), θ 0. a. Find the PDF of...
Let Xi, , Xn be a sample from U(0,0), θ 0. a. Find the PDF of X(n). b. Use Factorization theorem to show that X(n) is sufficient for θ. C. Use the definition of complete statistic to verify that X(n) is complete for θ.
6. Suppose that Xi,X2, X, is a random sample from the uniform distribution on (0,1). Let X(i), i ...
6. Suppose that Xi,X2, X, is a random sample from the uniform distribution on (0,1). Let X(i), i = 1, , n denote the order statistics. (a) Obtain the joint distribution of R- X)-X) and SXXn/2 b) Obtain the marginal pdf of S.
6. Suppose that Xi,X2, X, is a random sample from the uniform distribution on (0,1). Let X(i), i = 1, , n denote the order statistics. (a) Obtain the joint distribution of R- X)-X) and SXXn/2 b)...
+1,20] distribution, where , X.) 3. Let Xi,...,X be a random sample from Uniform θ >...
+1,20] distribution, where , X.) 3. Let Xi,...,X be a random sample from Uniform θ > 1 is unknown. Let X(1)-min(X, , X.) and X() = max(X,, (a) Derive the edf of X(n) and then its pdf. b) Derive EoX(n) (c) Find a function g(X(n)) such that Eolg(X())-θ for all θ > 1. (d) Replace X(n) by Xu) in the above questions, parts (a) - (c), and answer those.
Again, let X1,..., Xn be iid observations from the Uniform(0,0) distribution. (a) Find the joint pdf of Xi) and X(n)- (b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (c) It turns out, if Xi, . . . , xn (iid) Uniform(0,0), E(R)-θ . What happens to E® as n increases? Briefly explain in words why this makes sense intuitively.
Ques 3
(d) Suppose that n-10, and Xi Xio represent the waiting times that the 10 people must wait at a bus stop for their bus to arrive. Interpret the result of (c) in the context of this scenario be iid observations from the Uniform(0,0) distribution. 3. Again, let X..., X (a) Find the joint pdf of Xu) and X() (b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (e) It turns out, if X...
Only ques 4
(b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (c) It turns out, if Xi, . . . , Xn ~ (iid) Uniform(0,0), E(R)-θ . What happens to E(R) as n increases? Briefly explain in words why this makes sense intuitively. 4. Let X. Xn be a random sample from a population with pdf xotherwise Let Xa)<..< X(n) be the order statistics. Show that Xa)/X() and X(n) are independent random variables 5....
Problem 8: 5 points] Let Xi,.,.Xn be IID from a Uniform distribution on (-0,0) where 0 0 is an unknown parameter (a) Find a minimal sufficient statistic T. (b) Define Show that T and V are independent.
PROBLEM 3 Let X1, X2,L , X, be iid observations from a distribution with pdf given by f(xl0)=0x0-, 0<x<1, 0<O<00. a) Find the maximum likelihood estimator of O. b) Find the moment estimator of 0. c) (Extra credit) Compare the mean squared error of the two estimators in (a) and (b). Which one is better? (5 points)
Suppose X1, X2, ..., Xn are independent and identically distributed (iid) with a Uniform -0,0 distri- bution for some unknown e > 0, i.e., the Xi's have pdf Suppose X1, X2,..., Xn are independent and identically distributed (iid f(3) = S 20, if –0 < x < 0; 20 0, otherwise. (a) (4 pts) Briefly explain why or why not this is an exponential family (b) (5 pts) Find one meaningful sufficient statistic for 0. (By "meaningful”, I mean it...
1. Let X be an iid sample of size n from a continuous distribution with mean /i, variance a2 and such that Xi e [0, 1] for all i e {1,...,n}. Let X = average. For a E (0,1), we wish to obtain a number q > 0 such that: (1/n) Xi be the sample Р(X € |и — 9. и + q) predict with probability approximately In other words, we wish to sample of size n, the average X...
Let Xi, , Xn be a sample from U(0,0), θ 0. a. Find the PDF of X(n). b. Use Factorization theorem to show that X(n) is sufficient for θ. C. Use the definition of complete statistic to verify that X(n) is complete for θ.
6. Suppose that Xi,X2, X, is a random sample from the uniform distribution on (0,1). Let X(i), i = 1, , n denote the order statistics. (a) Obtain the joint distribution of R- X)-X) and SXXn/2 b) Obtain the marginal pdf of S.
6. Suppose that Xi,X2, X, is a random sample from the uniform distribution on (0,1). Let X(i), i = 1, , n denote the order statistics. (a) Obtain the joint distribution of R- X)-X) and SXXn/2 b)...
+1,20] distribution, where , X.) 3. Let Xi,...,X be a random sample from Uniform θ > 1 is unknown. Let X(1)-min(X, , X.) and X() = max(X,, (a) Derive the edf of X(n) and then its pdf. b) Derive EoX(n) (c) Find a function g(X(n)) such that Eolg(X())-θ for all θ > 1. (d) Replace X(n) by Xu) in the above questions, parts (a) - (c), and answer those.
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