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- b) Let X₁, X₂,..., X, denote a random sample of size n from a distribution with probability density function: i) ii) iii) f(x; 0) = {0(1-x)(¹+0), Find the maximum likelihood estimator of 0. Derive fisher's information of 0. suppose that 5 independent observations were taken from such a distribution: 4 1 3 5 4 and 0 = 0.7052 Find a 99% confidence interval for 0. x>0 elsewhere Da) Let X₁, X₂, X3, X30 be a random sample of size 30 from a population distributed with the following probability density function: f(x)=2 if 0Let 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 -∞07)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: X~ f (x I θ) . Find the distribution function of the random variable X.b) 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.b) Let X₁, X2, X3.....Xn be a random sample of n from population X distributed with the following probability density function: ze zo, f(x;0)=√2m0 0, (i) Find the parameter space of 0. (ii) Find the maximum likelihood estimator of 0. if -∞b) Let X₁, X2, X3,..., X30 be a random sample of size 30 from the population X with the moment generating function given by Mx (t) = ez(et-1) Suppose that Y = Σ201 X₁, i) Determine the probability density function of Y and find P(6 < Y < 10). ii) Use the central limit theorem to compute P(6b) Let X₁, X2, X3,...,Xn be a random sample of n from population X distributed with the following probability density function: f(x; 0) = 1 -20₁ V2πθ e 0, 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.b) Let Y1, Y2.,Y5 be independent random variables with probability density function y 1 e 4 4 f(y)= ,y > 0 ,elsewhere 3 Determine the distribution and parameter of V = EY; using the method of moment i=1 generating function. Hence, find the mean of V using the moment generating function.Suppose that X and Y are statistically independent and identically distributed uniform random variables on (0,1). (a) Write down the joint probability density function fx,y(x,y) of X and Y on its support.Let Y₁, Y₂,..., Yn be a random sample from the inverse Gaussian distribution with probability density function: f(y, μ, 2) = { Where μ and are unknown. a) What are i. 1 -ACT √ λ 2ny3)že elsewhere if y > 0 the likelihood ii. the log likelihood functions of u and λ. b) Find the maximum likelihood estimators of u and 2.12. Let the random variable X and Y have joint pdf 4 f(x,y) = (x² + 3y²), 0Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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