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Problem 1: x2 Expectation - The Hard Way The pdf of a x distribution is given...
Exercise 3 For the computation of the expectation Ef[h(x)] when f is the normal pdf and h(x) - ex...
Exercise 3 For the computation of the expectation Ef[h(x)] when f is the normal pdf and h(x) - exp A. Show that Ef[h(x)] can be computed in closed form and derive its value. ( 2p) + exp (-(xur) -(x-2)2 (x-4)2 Construct a regular Monte Carlo approximation based on a normal N (0,1) sample of size Nsim-10A4 and produce an error evaluation. Compare the above with an importance sampling approximation based on an importance function g corresponding to the U(-6 -...
Consider the simplified Bayesian model for normal data The joint posterior pdf is ful, σ21 x)a(σ2,-/2-1 expl_jy.tx, _aP...
Consider the simplified Bayesian model for normal data The joint posterior pdf is ful, σ21 x)a(σ2,-/2-1 expl_jy.tx, _aPI The marginal posterior pdfs of μ and σ 2 can be obtained by integrating out the other variable (8.30) y@1 x) α (σ2)-m;,-1/2 expl-- Σ.-tri-x)2 (8.31) d. Let q1 and q2 be they/2 and 1-y/2 quantiles of (8.31). Show that the 1-γ credible interval (gi,q2) is identical to the classic confidence interval (5.19) (with ar replaced by y). Hence, a (1-α) stochastic...
Let X1,.. ,X be a random sample from an N(p,02) distribution, where both and o are...
Let X1,.. ,X be a random sample from an N(p,02) distribution, where both and o are unknown. You will use the following facts for this ques- tion: Fact 1: The N(u,) pdf is J(rp. σ)- exp Fact 2 If X,x, is a random sample from a distribution with pdf of the form I-8, f( 0,0) = for specified fo, then we call and 82 > 0 location-scale parameters and (6,-0)/ is a pivotal quantity for 8, where 6, and ô,...
Let X1, X2, ..., Xn be a random sample from the distribution with pdf f(3;6) =...
Let X1, X2, ..., Xn be a random sample from the distribution with pdf f(3;6) = V porta exp ( 0) 10.02) for some parameter 2 > 0. (a) Find the MLE for 0. (b) Find the Cramér-Rao lower bound for the variance of all unbiased estimators of 0. (c) Find the asymptotic distribution of your MLE from part (a).
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)
Let X1,X be a random sample from an EXP(0) distribution (0 > 0) You will use the following facts for this question:...
Let X1,X be a random sample from an EXP(0) distribution (0 > 0) You will use the following facts for this question: Fact 1: If X EXP(0) then 2X/0~x(2). Fact 2: If V V, are a random sample from a x2(k) distribution then V V (nk) (a) Suppose that we wish to test Ho : 0 against H : 0 = 0, where 01 is specified and 0, > Oo. Show that the likelihood ratio statistic AE, O0,0)f(E)/ f (x;0,)...
That is, the distribution of X has pdf given by θ-11(1 < x 0) and a...
That is, the distribution of X has pdf given by θ-11(1 < x 0) and a point mass on {x-1). (b) Let X1, X2,..., Xn be a random sample from the distribution in part (a). Show that the pdf of the maximum order statistic X(n) is given by JX(n) (c) Show that X(n) is a sufficient statistic for θ. Is X(n) complete?
3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist of Y=F(X) b) Show that Z=-21n(Y) has a Gamma dist. & derive it. 4. X-i ~ cont with pdf fi(x) and CDF Fi(x), i=1, 2, , k....
3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist of Y=F(X) b) Show that Z=-21n(Y) has a Gamma dist. & derive it. 4. X-i ~ cont with pdf fi(x) and CDF Fi(x), i=1, 2, , k. all independent. Define YjaFi(Xi), i=1, , k. Derive the distribution of
3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist of Y=F(X) b) Show that Z=-21n(Y) has a Gamma dist....
2. X has pdf fx (+) = 3x I(0 <r <1) and Y has conditional distribution, given X = r, of Uniform(-1,2). a) Obt...
2. X has pdf fx (+) = 3x I(0 <r <1) and Y has conditional distribution, given X = r, of Uniform(-1,2). a) Obtain the pdf of X, Y. Sketch the support of this pdf. b) Obtain E(Y|X) and E(YPX). Also obtain E(XY|X) by using an appropriate property of conditional expectation and one of the previous two calculations c) Find Cov(X,Y), that is the covariance of X with Y. Are X and Y independent? Justify your answer. The next page...
Problem 4 (Conditional Expectation and Variance). Suppose the joint distri- bution of (X, Y) is given...
Problem 4 (Conditional Expectation and Variance). Suppose the joint distri- bution of (X, Y) is given by the following contingency (row represents x) table 20 points (x,y) 2 4 6 1 0.3 0 0.1 2 0 0.2 0 3 0.1 0 0.3 A) Compute the marginal distributions of X and Y B) Are X and Y independent? Explain. C) Find the conditional distribution of Y given X -1 D) Compute E[Y|X 1] E) Compute EY|X= 2] F Compute E[exp(X)Y|x 2
Exercise 3 For the computation of the expectation Ef[h(x)] when f is the normal pdf and h(x) - exp A. Show that Ef[h(x)] can be computed in closed form and derive its value. ( 2p) + exp (-(xur) -(x-2)2 (x-4)2 Construct a regular Monte Carlo approximation based on a normal N (0,1) sample of size Nsim-10A4 and produce an error evaluation. Compare the above with an importance sampling approximation based on an importance function g corresponding to the U(-6 -...
Consider the simplified Bayesian model for normal data The joint posterior pdf is ful, σ21 x)a(σ2,-/2-1 expl_jy.tx, _aPI The marginal posterior pdfs of μ and σ 2 can be obtained by integrating out the other variable (8.30) y@1 x) α (σ2)-m;,-1/2 expl-- Σ.-tri-x)2 (8.31) d. Let q1 and q2 be they/2 and 1-y/2 quantiles of (8.31). Show that the 1-γ credible interval (gi,q2) is identical to the classic confidence interval (5.19) (with ar replaced by y). Hence, a (1-α) stochastic...
Let X1,.. ,X be a random sample from an N(p,02) distribution, where both and o are unknown. You will use the following facts for this ques- tion: Fact 1: The N(u,) pdf is J(rp. σ)- exp Fact 2 If X,x, is a random sample from a distribution with pdf of the form I-8, f( 0,0) = for specified fo, then we call and 82 > 0 location-scale parameters and (6,-0)/ is a pivotal quantity for 8, where 6, and ô,...
Let X1, X2, ..., Xn be a random sample from the distribution with pdf f(3;6) = V porta exp ( 0) 10.02) for some parameter 2 > 0. (a) Find the MLE for 0. (b) Find the Cramér-Rao lower bound for the variance of all unbiased estimators of 0. (c) Find the asymptotic distribution of your MLE from part (a).
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)
Let X1,X be a random sample from an EXP(0) distribution (0 > 0) You will use the following facts for this question: Fact 1: If X EXP(0) then 2X/0~x(2). Fact 2: If V V, are a random sample from a x2(k) distribution then V V (nk) (a) Suppose that we wish to test Ho : 0 against H : 0 = 0, where 01 is specified and 0, > Oo. Show that the likelihood ratio statistic AE, O0,0)f(E)/ f (x;0,)...
That is, the distribution of X has pdf given by θ-11(1 < x 0) and a point mass on {x-1). (b) Let X1, X2,..., Xn be a random sample from the distribution in part (a). Show that the pdf of the maximum order statistic X(n) is given by JX(n) (c) Show that X(n) is a sufficient statistic for θ. Is X(n) complete?
3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist of Y=F(X) b) Show that Z=-21n(Y) has a Gamma dist. & derive it. 4. X-i ~ cont with pdf fi(x) and CDF Fi(x), i=1, 2, , k. all independent. Define YjaFi(Xi), i=1, , k. Derive the distribution of
3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist of Y=F(X) b) Show that Z=-21n(Y) has a Gamma dist....
2. X has pdf fx (+) = 3x I(0 <r <1) and Y has conditional distribution, given X = r, of Uniform(-1,2). a) Obtain the pdf of X, Y. Sketch the support of this pdf. b) Obtain E(Y|X) and E(YPX). Also obtain E(XY|X) by using an appropriate property of conditional expectation and one of the previous two calculations c) Find Cov(X,Y), that is the covariance of X with Y. Are X and Y independent? Justify your answer. The next page...
Problem 4 (Conditional Expectation and Variance). Suppose the joint distri- bution of (X, Y) is given by the following contingency (row represents x) table 20 points (x,y) 2 4 6 1 0.3 0 0.1 2 0 0.2 0 3 0.1 0 0.3 A) Compute the marginal distributions of X and Y B) Are X and Y independent? Explain. C) Find the conditional distribution of Y given X -1 D) Compute E[Y|X 1] E) Compute EY|X= 2] F Compute E[exp(X)Y|x 2
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