Assume a sample X1，X2 ，，Xn are i.i.d. random variables from a distribution given below. (a) Assume that the sample is from a distribution with density function f(x |σ) = 2/1/2 exp (-2¹). Note that here | x | denotes the absolute value of x.o)20σPlease find the maximum likelihood estimate of o.(b) Assume that the sample is from the Pareto distribution, which has been usedin economics as a model for a density function with a slowly decaying tail.The density function is given by f(x | xo, 0)0x8x-0-1хох* > £0,0 > 1.Here xo> 0 is given. Please find the maximum likelihood estimate of 0.=9

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Answered: (a) Assume that the sample is from a…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 25EQ

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Let X1, X2,...,X, be a random sample from a distribution with density function e if x > 0 f(x; 0) elsewhere What is the maximum likelihood estimator of 0 ?
Let X,, X2,... ,X, be a random sample from a distribution with density function if x20 f(x; 0) elsewhere What is the maximum likelihood estimator of 0 ?
Consider fitting a continuous distribution with the density f (x) exp (-0|x|) 2 for - 0 < x < ∞ The mean of this distribution is 0 and the variance is 20-2. 0 is the unknown parameter. Say you observe data -1.7,1.2,-0.3,1.6,-1.3,0.1 a) Find the maximum likelihood estimate for 0. b) Explain why you cannot use the method of moments to find an estimate by matching the sample mean to the population mean c) Use method of moments in some other way to estimate 0. d) Say instead you received the data 1.7,4.5,1.2,0.3,1.6,1.3,0.1,1.6,2.5? Does this distri- bution appear to be a good fit for the data?
The life length, T, of an electric component is a random variable with hazard rate function h(t) = (60+t) -;7>0. Obtain (i) Cumulative Hazard Rate Function, (ii) Reliability Function, (iii) Density Function and (iv) Conditional Reliability Function for T.
Random variables Y₁,... Yn follow the probability density distribution: f(y|0) = { v with known expected values E[Y] = 20 and Var[Y] = 202. a.) Find the maximum likelihood estimate of 0. b.) Find E[] and V(A) y >0 otherwise c.) Is this a biased estimator?
Let the random variable X be defined on the support set (1,2) with pdf fX(x) = (4/15)x3, Find the variance of X.
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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Borrowers leave the issue desk of a small library according to a Poisson process with rate 40 per hour. The number of books issued to a borrower has a uniform distribution with probability mass function x=1,2,..., 6. px(x)= (a) Calculate the mean and variance of the number of books issued by the library to a borrower. The random variable S(t) denotes the total number of books issued in the time interval (0, t], where t is measured in hours. (b) Calculate the mean and variance of the number of books issued in a morning (9 am-12 noon). (c) Calculate the index of dispersion for the random process {S(t); t >0}, and comment on what your value tells you about the pattern of book-borrowing at the library.

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