Give an example of a sequence of iid random variables,X1,X2,⋯ for which condition imnnP(|X1|>n)=0 does not...
Give an example of a sequence of iid random variables,X1,X2,⋯
for which condition imnnP(|X1|>n)=0 does not hold and the WLLN
of Khintchin for iid case fails.
The following ``answers'' have been proposed. Please read the
choices very carefully and pick the most complete and accurate
choice.
(a) Take X1,X2,⋯so that each one of the random variables has Pareto
density. That is, the density of X1 is f(t)=ct2or |t|>1 and zero
otherwise, where c is a constant so that ∫ℜf(t)dt=1.
(b) Take X1,X2,⋯ so that each one of the random variables has the
Cauchy density. That is, the density of X1 is f(t)=1π(1+t2)for
t∈ℜ
(c) Both (a) and (b) are valid examples..
(d) Only (b) is a valid example.
(e) None of the above
Homework Answers
Add Answer to:
Give an example of a sequence of iid random variables,X1,X2,⋯
for which condition imnnP(|X1|>n)=0 does not...
(3) Let XXnX1,X2,⋯,Xn be iidiid random variables with Cauchy(0,1)Cauchy(0,1) distribution. That is, the density of X1...
(3) Let XXnX1,X2,⋯,Xn be iidiid random variables with Cauchy(0,1)Cauchy(0,1) distribution. That is, the density of X1 is 1/(π(1+x2)) for x∈ℜ. Prove that (X1+X2+⋯+Xn)/n is again distributed as Cauchy(0,1). The following ``answers'' have been proposed. Please read the choices very carefully and pick the most complete and accurate choice. (a) By the last exercise, the characteristic function of X1, is e−|t|e−|t|. Therefore by the fact that the Xi are iid, the characteristic function of their average is the product of n...
two random variables x1 and x2 have a joint probability density function f(x1,x2)={x1+x2, 0<x1<1, 0<x2<1 0,...
two random variables x1 and x2 have a joint probability density function f(x1,x2)={x1+x2, 0<x1<1, 0<x2<1 0, otherwise what is the marginal distribution of x1 and x2
Suppose X1 and X2 are iid Poisson(θ) random variables and let T = X1 + 2X2....
Suppose X1 and X2 are iid Poisson(θ) random variables and let T = X1 + 2X2. (a) Find the conditional distribution of (X1,X2) given T = 7. (b) For θ = 1 and θ = 2, respectively, calculate all probabilities in the above conditional distribution and present the two conditional distributions numerically.
Let X1,X2,...,Xn be iid N(μ,1) random variables. Find the MVUE of θ=μ2.
Let X1,X2,...,Xn be iid N(μ,1) random variables. Find the MVUE of θ=μ2.
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and...
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
Let X1, X2, ..., X, be iid random variables with a "Rayleigh” density having the following...
Let X1, X2, ..., X, be iid random variables with a "Rayleigh” density having the following pdf: f(x) = 2x2=+*10, 2 > 0 > 0 V лв a) (3 points) Find a sufficient estimator for using the Factorization Theorem. b) (3 points) Find a method of moments estimator for 0. Small help: E(X1) = c) (7 points) What is the MLE of 02 +0 - 10 ? d) (7 points) For a fact, IX has a Gamman, o) distribution. Using...
Suppose that X1,X2,. X are iid random variables with pdf ,220 (a) Find the maximum likelihood...
Suppose that X1,X2,. X are iid random variables with pdf ,220 (a) Find the maximum likelihood estimate of the parameter a (b) Find the Fisher Information of X1,X2,.. ., Xn and use it to estimate a 95% confidence interval on the MLE of a (c) Explain how the central limit theorem relates to (b).
T (0, c(X1-X2)2) įs a Let X, and X2 be iid. N(0, (Au)100% confidence interval for σ- 1) σ2) varia...
t (0, c(X1-X2)2) įs a Let X, and X2 be iid. N(0, (Au)100% confidence interval for σ- 1) σ2) variables) . Find a constant so tha
t (0, c(X1-X2)2) įs a Let X, and X2 be iid. N(0, (Au)100% confidence interval for σ- 1) σ2) variables) . Find a constant so tha
Question 1: Suppose that X1, X2,... Xn are independent identically distributed continuous outcome random variables which...
Question 1: Suppose that X1, X2,... Xn are independent identically distributed continuous outcome random variables which have a probability density function (pdf) f(z) = π1+ア Calculate (with all working) the pdf of the average of the X,i Comment on the significance of this result to sampling from a random vari- able with the pdf f. This pdf is called a Cauchy density.
Suppose that X1, X2, ..., Xn are independent random variables (not iid) with densities ÍXi(z10,) -.2...
Suppose that X1, X2, ..., Xn are independent random variables (not iid) with densities ÍXi(z10,) -.2 e _ θ:/z1(z > 0), where θί 〉 0, for i = 1, 2, , n. (a) Derive the form of the likelihood ratio test (LRT) statistic for testing versuS H1: not Ho. You do not have to find the distribution of the likelihood ratio test (LRT) statistic under Ho- Just find the form of the statistic. (b) From your result in part (a),...
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
Let X1, X2, ..., X, be iid random variables with a "Rayleigh” density having the following pdf: f(x) = 2x2=+*10, 2 > 0 > 0 V лв a) (3 points) Find a sufficient estimator for using the Factorization Theorem. b) (3 points) Find a method of moments estimator for 0. Small help: E(X1) = c) (7 points) What is the MLE of 02 +0 - 10 ? d) (7 points) For a fact, IX has a Gamman, o) distribution. Using...
Suppose that X1,X2,. X are iid random variables with pdf ,220 (a) Find the maximum likelihood estimate of the parameter a (b) Find the Fisher Information of X1,X2,.. ., Xn and use it to estimate a 95% confidence interval on the MLE of a (c) Explain how the central limit theorem relates to (b).
t (0, c(X1-X2)2) įs a Let X, and X2 be iid. N(0, (Au)100% confidence interval for σ- 1) σ2) variables) . Find a constant so tha
t (0, c(X1-X2)2) įs a Let X, and X2 be iid. N(0, (Au)100% confidence interval for σ- 1) σ2) variables) . Find a constant so tha
Question 1: Suppose that X1, X2,... Xn are independent identically distributed continuous outcome random variables which have a probability density function (pdf) f(z) = π1+ア Calculate (with all working) the pdf of the average of the X,i Comment on the significance of this result to sampling from a random vari- able with the pdf f. This pdf is called a Cauchy density.
Suppose that X1, X2, ..., Xn are independent random variables (not iid) with densities ÍXi(z10,) -.2 e _ θ:/z1(z > 0), where θί 〉 0, for i = 1, 2, , n. (a) Derive the form of the likelihood ratio test (LRT) statistic for testing versuS H1: not Ho. You do not have to find the distribution of the likelihood ratio test (LRT) statistic under Ho- Just find the form of the statistic. (b) From your result in part (a),...
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