Answered: ) Let X, X2, ..., X, be a random sample… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 31E

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) Let X,, X,, ..., X, be a random sample
from the N(0, o²)-distribution, and suppose that the prior distribution for 0 is the N(H, 7²)-
distribution, where o, µ and 7’ are known. Determine the posterior distribution for 6,
given X,. X2, .... X,, and the optimal estimator of 0 under (i) quadratic loss, and
(ii) absolute error loss.

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A 99 percent one-sample z-interval for a proportion will be created from the point estimate obtained from each of two random samples selected from the same population: sample R and sample S. Let R represent a random sample of size 1,000, and let S represent a random sample of size 4,000. If the point estimate obtained from R is equal to the point estimate obtained from S, which of the following must be true about the respective margins of error constructed from those samples? (A) The margin of error for S will be 4 times the margin of error for R. (B) The margin of error for S will be 2 times the margin of error for R. (C) The margin of error for S will be equal to the margin of error for R. (D) The margin of error for R will be 4 times the margin of error for S. (E) The margin of error for R will be 2 times the margin of error for S.
Let X1, X2, ..., X10 be a random sample from a gamma distribution with α = 3 and β = 1/θ. Suppose we believe that theta follows a gamma-distribution with α = 28 and β = 17 and suppose we have a trial (X1 , ... , Xn ) with an observed x̄ = 28.2 What is the posterior distribution of θ What is the Bayes point estimate of θ associated with the square-error loss function?
Let X1, X2, ..., X10 be a random sample from a gamma distribution with α = 3 and β = 1/θ. Suppose we believe that theta follows a gamma-distribution with α = 28 and β = 17 and suppose we have a trial (X1 , ... , Xn ) with an observed x̄ = 28.2 What is the posterior distribution of θ What is the Bayes point estimate of θ associated with the square-error loss function? What is the Bayes point estimate of θ using the mode of the posterior distribution?
Let X₁,X₂,...,Xₙ denote a random sample from a distribution that is N(0,θ), where the variance θ is an unknown positive number. Show that there exists a uniformly most powerful test of size α for testing the simple hypothesis H₀ : θ = θ', where θ' is a fixed positive number.
Let X1 and X2 be observations of a random sample of size n = 2 from a Cauchy Distribution.Find P(X1 < −1 and 1 < X2)
A 99 percent one-sample z-interval for a proportion will be created from the point estimate obtained from each oftwo random samples selected from the same population: sample R and sample S. Let R represent a random sampleof size 1,000, and let S represent a random sample of size 4,000. If the point estimate obtained from R is equal tothe point estimate obtained from S, which of the following must be true about the respective margins of errorconstructed from those samples?(A) The margin of error for S will be 4 times the margin of error for R.(B) The margin of error for S will be 2 times the margin of error for R.(C) The margin of error for S will be equal to the margin of error for R.(D) The margin of error for R will be 4 times the margin of error for S.(E) The margin of error for R will be 2 times the margin of error for S.
Let Z be a N(0, 1) random variable, and let F(x) be thecumulative distribution function for Z. Show that on S (∞, 0], F(x) is an increasing convex function, and on S [0, ∞), F(x) is an increasing concave function.
Suppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on interval [0, 1]. Determine the value of E[(X1-2X2+X3)2]
(a) Let X1, X2, X3, X4 be IID uniform on (0,1) random variables and let S4 = X1 +X2 +X3 +X4. Use normal approximation to calculate P(S4 ≥ 3) approximately by linear interpolation. b)Suppose the IQ scores of a million individuals have a mean of 100 and SD of 10. Without making any further assumptions about the distribution of the scores, find an upper bound on the number of scores exceeding 130.
If the random variable X follows the uniform distribution U= (0,1) What is the distribution of the random variable Y= -2lnX. Show its limits.
Suppose that the random variables X1,...,Xn form a random sample of size n from the uniform distribution on the interval [0, 1]. Let Y1 = min{X1,. . .,Xn}, and let Yn = max{X1,...,Xn}. Find E(Y1) and E(Yn).
Let X1, X2, ... , Xn be a random sample, normally distributed with mean μ and variance σ2If σ2 is unknown, find a minimum value for n to guarantee, with probability 0.90, that a 0.95 CI for μ will have length no more than σ/4

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