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Homework Answers
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2 Let X1, X2, ...,X, be independent continuous random variables from the following distribution: f(3) =...
2 Let X1, X2, ...,X, be independent continuous random variables from the following distribution: f(3) = ox-(0-1) where : > 1 and a > 1 You may use the fact: E[X]- .- 2.1 Show that the maximum likelihood estimator of a is ômle = Ei log Xi 2.3 Derive a sufficient statistic for a. What theorem are you using to determine sufficiency? 2.4 Show that the fisher information in the whole sample is: 1(a)= 2.5 What Cramer Rao lower bound...
Let X1, ..., X, be a random sample with pdf defined as: f(x) = 2x exp{...
Let X1, ..., X, be a random sample with pdf defined as: f(x) = 2x exp{ –x?/0}, where > 0. The distribution of the MLE is: O None of the alternatives. o ên ~ Gamman,/n) Oô - Gammale, n/=) Oô - Exp(0) O 6 – Exp(9/1)
(1 point) Let X1 and X2 be a random sample of size n= 2 from the...
(1 point) Let X1 and X2 be a random sample of size n= 2 from the exponential distribution with p.d.f. f(x) = 4e - 4x 0 < x < 0. Find the following: a) P(0.5 < X1 < 1.1,0.3 < X2 < 1.7) = b) E(X1(X2 – 0.5)2) =
8. Let X1, X2,...,X, U(0,1) random variables and let M = max(X1, X2,...,xn). - Show that...
8. Let X1, X2,...,X, U(0,1) random variables and let M = max(X1, X2,...,xn). - Show that M. 1, that is, M, converges in probability to 1 as n o . - Show that n(1 - M.) Exp(1), that is, n(1 - M.) converges in distribution to an exponential r.v. with mean 1 as n .
6.1.10. Let X1, X2..... Xn be a random sample from a N(0,0%) distribution, where o? is...
6.1.10. Let X1, X2..... Xn be a random sample from a N(0,0%) distribution, where o? is fixed but-X <O<O. (a) Show that the mle ofis X. (b) If is restricted by 0 < < oc, show that the mie of 8 is 8 = max{0,X}.
estimator of 3. (14 points each) Let X1, X2,..., X, be a random sample from Gammala,...
estimator of 3. (14 points each) Let X1, X2,..., X, be a random sample from Gammala, 1) distribution where a is known, and is unknown. (i) Find the moment estimator of X. (ii) Find the MLE of i noints each Let X1, X ., X, be a sample from N(u,0%).
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a)...
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3).
3. (25 pts.) Let X1,...
3. Let X1, X2, . . . , Xn be a random sample from a distribution...
3. Let X1, X2, . . . , Xn be a random sample from a distribution with the probability density function f(x; θ) (1/02)Te-x/θ. O < _T < OO, 0 < θ < 00 . Find the MLE θ
Let X1, X2,...,Xn be a random sample from the exponential distribution with rate A Let c > 0 be a fixed and known nu...
Let X1, X2,...,Xn be a random sample from the exponential distribution with rate A Let c > 0 be a fixed and known number. For i 1,2 п, let ..1 -{: : if Xic 1 Y otherwise Suppose that you get to observe Yı, Y2,... , Y,n but you do not get to observe X1, X2,... , X,n п. Find the MLE for X based on this information
May 21, 2019 R 3+3+5-11 points) (a) Let X1,X2, . . Xn be a random sample from G distribution. Show that T(Xi, . . . , x,)-IT-i xi is a sufficient statistic for a (Justify your work). (b) Is Unifo...
May 21, 2019 R 3+3+5-11 points) (a) Let X1,X2, . . Xn be a random sample from G distribution. Show that T(Xi, . . . , x,)-IT-i xi is a sufficient statistic for a (Justify your work). (b) Is Uniform(0,0) a complete family? Explain why or why not (Justify your work) (c) Let X1, X2, . .., Xn denote a random sample of size n >1 from Exponential(A). Prove that (n - 1)/1X, is the MVUE of A. (Show steps.)....
2 Let X1, X2, ...,X, be independent continuous random variables from the following distribution: f(3) = ox-(0-1) where : > 1 and a > 1 You may use the fact: E[X]- .- 2.1 Show that the maximum likelihood estimator of a is ômle = Ei log Xi 2.3 Derive a sufficient statistic for a. What theorem are you using to determine sufficiency? 2.4 Show that the fisher information in the whole sample is: 1(a)= 2.5 What Cramer Rao lower bound...
Let X1, ..., X, be a random sample with pdf defined as: f(x) = 2x exp{ –x?/0}, where > 0. The distribution of the MLE is: O None of the alternatives. o ên ~ Gamman,/n) Oô - Gammale, n/=) Oô - Exp(0) O 6 – Exp(9/1)
(1 point) Let X1 and X2 be a random sample of size n= 2 from the exponential distribution with p.d.f. f(x) = 4e - 4x 0 < x < 0. Find the following: a) P(0.5 < X1 < 1.1,0.3 < X2 < 1.7) = b) E(X1(X2 – 0.5)2) =
8. Let X1, X2,...,X, U(0,1) random variables and let M = max(X1, X2,...,xn). - Show that M. 1, that is, M, converges in probability to 1 as n o . - Show that n(1 - M.) Exp(1), that is, n(1 - M.) converges in distribution to an exponential r.v. with mean 1 as n .
6.1.10. Let X1, X2..... Xn be a random sample from a N(0,0%) distribution, where o? is fixed but-X <O<O. (a) Show that the mle ofis X. (b) If is restricted by 0 < < oc, show that the mie of 8 is 8 = max{0,X}.
estimator of 3. (14 points each) Let X1, X2,..., X, be a random sample from Gammala, 1) distribution where a is known, and is unknown. (i) Find the moment estimator of X. (ii) Find the MLE of i noints each Let X1, X ., X, be a sample from N(u,0%).
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3).
3. (25 pts.) Let X1,...
3. Let X1, X2, . . . , Xn be a random sample from a distribution with the probability density function f(x; θ) (1/02)Te-x/θ. O < _T < OO, 0 < θ < 00 . Find the MLE θ
Let X1, X2,...,Xn be a random sample from the exponential distribution with rate A Let c > 0 be a fixed and known number. For i 1,2 п, let ..1 -{: : if Xic 1 Y otherwise Suppose that you get to observe Yı, Y2,... , Y,n but you do not get to observe X1, X2,... , X,n п. Find the MLE for X based on this information
May 21, 2019 R 3+3+5-11 points) (a) Let X1,X2, . . Xn be a random sample from G distribution. Show that T(Xi, . . . , x,)-IT-i xi is a sufficient statistic for a (Justify your work). (b) Is Uniform(0,0) a complete family? Explain why or why not (Justify your work) (c) Let X1, X2, . .., Xn denote a random sample of size n >1 from Exponential(A). Prove that (n - 1)/1X, is the MVUE of A. (Show steps.)....
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