1) Let X₁, X2, X3, ... Xn be a sequence of independent and identically distributed with thefollowing probability density functionf(x; 0) = {(0 + ¹S(0+1)x-0-2if 1 < x < 00otherwiseb) Use method of moments to find a statistic that we can use to estimate theparameter 8.c) Check if Y = is an unbiased estimator of 0 or not, where ☎ is the estimatorobtained in b).d)Find the maximum likelihood estimator of the parameter 0.e) What is the fisher information in the sample of size n about the parameter 0.

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Answered: 1) Let X₁, X2, X3, ... Xn be a sequence…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Suppose that we have an independent and identically distributed sample from some distribution family G defined by an unknown parameter value 0. The frequentist approach to statistics would treat e as a fixed value (constant), but the Bayesian approach to statistics would treat 0 as a random variable with some prior distribution specified by a probability density function 1(0). True O False
Estimate the unknown parameter from a sample 3, 3, 3, 3, 3, 3, 3, 7, 7, 7, 7, 7 drawn from a discrete distribution with the probability mass functiontion 0 (P(3) P(7) = 1-0 0 - Compute the estimator of 0 use the method of moments.
Suppose that X is a random variable having the following probability density function given by (2(0-x) x>0 f(x) = elsewhere Find the value of c such that an interval from x to cx is a (1-a)100% confidence interval for the parameter 8.
Suppose that a random variable X has the following probability density function. SC (64-x²) 0 ≤ x ≤ 8 otherwise f(x) = {C Find the expected value of X. (You will need to find the value of the constant C so that f is a pdf.)
b) Let X to be a random sample from a distribution with a probability density function given by f(x; 0) = {0x0-¹,if 0 i) Find the level of significance of the test, a. ii) Calculate the power of the test.
Suppose random variable Y represents a random number generator that can take any value between 1 and 11. Then the density curve of the outcomes has constant height between 1 and 11, and height of 0 elsewhere. What’s the height of the density curve between 1 and 11? What is the probability that Y<2? What is the probability that Y=2? Can Y=2 happen as an event for this random number generator? What is the probability that Y=0? Can Y=0 happen as an event for this random number generator?
a) Suppose that X is a random variable having the following probability density function given by (2(0-x) 02 f(x) = -, x > 0 (0, elsewhere Find the value of c such that an interval from x to cx is a (1- a)100% confidence interval for the parameter 0.
A fair coin is flipped three times. Let X represent the number of heads to occur on the first two flips and Y the number of heads to occur on the last two flips. (a) Find the joint probability function, along with each marginal function, for X and Y. (b) Find cov(X, Y). (c) Are X and Y independent? Explain your answer. Hint: There are 8 outcomes for this expermient, e.g., if the outcome is HTT, then we have X = 1 and Y = 0.
b) Let Y,,Y2, .. , Yn denote a random sample from N(0,0) distribution with probability density function: f(y;8) = e V2n0 i) Show that f(y; 0) belongs to the 1-parameter exponential family. ii) What is the complete sufficient statistic for 0? Justify your answer. iii) Show whether or not, the maximum likelihood estimator is an unbiased estimator of 0. iv) Does the estimator attains the minimum variance unbiased estimator of 0.

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