iv)3X₁ +2X₂ + X3 and ₂ = 5x4 + X5 be an unbiased estimator. Which44estimator is more efficient?Let B₁= b)Let X₁, X2,...,Xn denote a random sample of size n taken from a distribution withprobability density function given by:ii)iii)f(x, ß) = -Estimate101120360-x5eXX>0Find an estimator for the parameter ß using the method of maximum likelihood.when the values are found to be24153675elsewhere4050 46Show that the estimator obtained in (i) is a minimum variance unbiased estimatorfor parameter ß.

Answered: Image /qna-images/answer/21f720de-0a03-48f5-b025-79d214f314be.jpg

Answered: Let X₁, X2,...,Xn denote a random…

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 X is a random variable with the density:f(x) = ( θxθ−1 if x ∈ (0, 1) 0 if otherwise The parameter θ > 0 is to be estimated. Suppose that x1 = 0.3, x2 = 0.6 are the sample data. Get the estimate of θ(a) using the Moment Method;(b) using the Maximum Likelihood Method.
1(a)Consider the random variable model with density function given byf(x)= { 2kx/45, 0≤x≤5 2k(30−x)/45 , 5≤x≤10 0, otherwise Compute the(i) expected value(ii) standard deviation.comment and discuss on the output of these two components with the type the random variable model defined (b). If f (x)=ce−0.2x , if 0≤x≤10 and 0, otherwiseBased on your computations below, explain and discuss the variousprobabilities with respect to the given exponential function.(i)deduce the constant c, (ii)compute the probabilities(alpha) p(x≥5) (beta) p(x<4) (gamma) p(3≤x≤11)
Suppose that X1, . . . , Xn is a random sample from the Normal distribution N (0, σ2 ) with parameter σ > 0, Find the Maximum Likelihood Estimation of σ.
If Y is a continuous, uniformly distributed random variable over the interval(4,10), then the value of the PDF between 4 and 10 is?
Let Y1, Y2, ... , Yn be a random sample of size n from a gamma distribution with parameters α = 1and β = 2. Derive the probability distribution of the sample mean Y̅ using moment-generatingfunctions.
If the random variable T is the time to failure of a commercial product and the values of its probability den-sity and distribution function at time t are f(t) and F(t), then its failure rate at time t is given by f(t)1 − F(t). Thus, thefailure rate at time t is the probability density of failure attime t given that failure does not occur prior to time t.(a) Show that if T has an exponential distribution, thefailure rate is constant. (b) Show that if T has a Weibull distribution (see Exer-cise 23), the failure rate is given by αβt β−1.
2)Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the maximum likelihood estimator (MLE) of parameter θ.
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.
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size nfrom a distribution with pdf f(x) = 1, 0 < x < 1, zero elsewhere. Show that thekth order statistic Yk has a beta pdf with parameters α = k and β = n − k + 1.
Consider two independent random variables X1 andX2 having the same Cauchy distributionf(x) = 1π(1 + x2)for − q < x < qFind the probability density of Y1 = X1 + X2 by usingTheorem 1 to determine the joint probability density ofX1 and Y1 and then integrating out x1. Also, identify thedistribution of Y1.
Let y1,y2,...,y10 be a random sample from an exponential pdf with unknown parameter λ. Find the form of the Generalized Likelihood Ratio Test for H0: λ = λ0 versus H1: λ doesn't equal λ0. What integral would have to be evaluated to determine the critical value if α were equal to 0.05?
Let Y1 < Y2 < Y3 be the order statistics of a random sample of size 3 froma distribution having the pdf f(x) = 2x, 0 < x < 1, zero elsewhere. Show thatZ1 = Y1/Y2, Z2 = Y2/Y3, and Z3 = Y3 are mutually independent.

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