8. Suppose that X₁, X₂, ..., X₂ is a random sample available from a population distributed as theBernoulli with parameter 0.(a) Find the Cramer-Rao lower bound variance for the estimator of 0.(b) Using the invariant property of maximum likelihood estimators,i. Find the estimator of 0² and show whether or not it is unbiased.ii. Find the Cramer-Rao lower bound variance for the estimator of 02 and its absoluteefficiency.

Given that X1, X2, . . ., Xn be a random sample from a population distributed as the Bernoulli with…

Answered: 8. Suppose that X₁, X2, ..., X₂ is a…

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 X, Y, Z are iid observations from a Poisson distribution with parameter λ, which is unknown. Consider the 3 estimators T1 = X + Y − Z, T2 = 2X + Y + Z 4 , T3 = 3X + Y + Z 5 . (a) Which among the above estimators are unbiased? (b) Among the class of unbiased estimators, which has the minimum variance?
Suppose that a random sample from a normal popula-tion with the known variance σ2 is to be used to test the null hypothesis μ = μ0 against the alternative hypothe-sis μ = μ1, where μ1 > μ0, and that the probabilities of type I and type II errors are to have the preassigned val-ues α and β. Show that the required size of the sample is given by n = σ2(zα + zβ)2(μ1 − μ0)2
If X is a uniformly distributed random varibale with a=9 and b=16, then Calculate the variance of X? Round to three decimal places
The management of a supermarket wants to adopt a new promotional policy of giving free gift to every customer who spends more than a certain amount per visit at this supermarket. The expectation of the management is that after this promotional policy is advertised, the expenditure for all customers at this supermarket will be normally distributed with mean 400 L.E and a variance of 900 L.E2. 1) If the management wants to give free gifts to at most 12% of the customers, what should the amount of expenditure be above which a customer would receive a free gift? 2) In a sample of 100 customers, what is the number of customers whose expenditure is between 420 L.E and 485 L.E? 3) What is a probability of selecting a customer whose expenditure is differ than the population mean expenditure by at most 50 L.E? 4) What is the percentage of customers whose expenditure is at least 450 L.E.?
The management of a supermarket wants to adopt a new promotional policy of giving free gift to every customer who spends more than a certain amount per visit at this supermarket. The expectation of the management is that after this promotional policy is advertised, the expenditure for all customers at this supermarket will be normally distributed with mean 400 L.E and a variance of 900 L.E2.1) If the management wants to give free gifts to at most 12% of the customers, what should the amount of expenditure be above which a customer would receive a free gift?
The records of a casualty insurance company show that, in the past, its clients have had a mean of 1.9 auto accidents per day with a variance of 0.0036. The actuaries of the company claim that the variance of thenumber of accidents per day is no longer equal to 0.0036. Suppose that we want to carry out a hypothesis test to see if there is support for the actuaries' claim. State the null hypothesis H0 and the alternative hypothesis H1 that we would use for this test.
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?
A snack food manufacturer estimates that the variance of the number of grams of carbohydrates in servings of its tortilla chips is 1.141.14. A dietician is asked to test this claim and finds that a random sample of 2424 servings has a variance of 0.950.95. At alpha equals 0.10α=0.10, is there enough evidence to reject the manufacturer's claim? Assume the population is normally distributed.
The records of a casualty insurance company show that, in the past, its clients have had a mean of 1.7 auto accidents per day with a variance of 0.0025 . The actuaries of the company claim that the variance of the number of accidents per day is no longer equal to 0.0025. Suppose that we want to carry out a hypothesis test to see if there is support for the actuaries' claim. State the null hypothesis H0 and the alternative hypothesis H1 that we would use for this test. H0 : ? H1 : ?
If X1 and X2 constitute a random sample of size n = 2from a Poisson population, show that the mean of thesample is a sufficient estimator of the parameter λ.
An alternative proof of Theorem 2 may be based on the fact that if X1, X2, ..., and Xn are independent ran-dom variables having the same Bernoulli distribution with the parameter θ, then Y = X1 + X2 +···+ Xn isa random variable having the binomial distribution withthe parameters n and θ.Verify directly (that is, without making use of the factthat the Bernoulli distribution is a special case of thebinomial distribution) that the mean and the variance ofthe Bernoulli distribution are μ = θ and σ2 = θ (1 − θ
Let X be a discrete random variable whose possible values are 1, 2, 3 with respective probabilities P1, P2, P3. If E[X]=2 and P2≥0.1, find the values of P1, P2, P3 that maximize Var(X). (Hint: What is the sum of P1, P2, and P3? Set up a system of equations to eliminate one of the variables. Then substitute into the variance formula to obtain an expression for the variance.)

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