Let X,...,X, be independent random variables, each with the following probability density function f(x) = 0 xº- for 00 is parameter to be estimated (i) Integrate the expectation of X, alnd derive a method of moments estimator for 0. (ii) Estimate the likelihood function, L(0; X) for 0, where X – (X,...,X,)'.
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- Solve for d)Simplify the likelihood ratio e (µ1,..., Hr, o2) max (µ1,...,Hr,o²)EO0 e (41,...,Hr; o2) ' max (41,...,Hr,0²)EO SSTR/(r-1) SSE/(nt-r) and show that A is a decreasing function of F =Let X and Y be independent Exp(2) random variables. Define W = X + Y. a) Determine the correlation between X and W. The joint pdf of X and W is: fx,w(x, w) = 1^(2) e^(-1w) I{O6) Allow a continuous probability distribution to be defined as f(x) = 4x³ for the range 0 sxs1. Demonstrate that the area underneath the curve is 1 (the function integrates to 1) and calculate the mean and variance of the probability distribution.Let X₁, X₂, X3,...,Xn be a random sample of n from population X distributed with the fol probability density function: 1 ze=20, f(x;0)=√√2n0 0, if -∞0b) Let X₁, X2, X3,...,Xn be a random sample of n from population X distributed with the following probability density function: f(x;0)=√√2n0 0, -20₁ if -∞0 < x <∞0 otherwise (i) Find the parameter space of 0. (ii) Find the maximum likelihood estimator of 0. (iii) Check whether or not the estimator obtained in (ii) is unbiased. (iv) Find the Fisher information in this sample of size n about the parameter 0.A continuous random variable X has a P.D.F. fx (x) = 3 X, 0Sr a) and (ii) P (x > b) = 0.05.The probability density function for the continuous random variable X is given by: (А(х? — 2х + 21) 0Let X1,...,X, be independent random variables with pdfs -{* 0 -i(0 – 1) 0. Find a two-dimensional sufficient statistic for 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.Let X ~ N(0,o²) and Y ~ N(0, v²) be independent random variables and Z = X +Y. (a) Find the LMMSE estimator of X given Z. (b) Find the conditional distribution fzix(z, x).Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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