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![Let the random variable X follow a uniform distribution in the interval [0,0]. Then the
density function of X is
f(x;0)
S, if 0<x<0
0, otherwise.
Consider a random sample of size n from this population.
(a) From the likelihood function of , argue that its maximum likelihood estimator is
the nth order statistic (Y).
(b) Is the estimator in part (a) unbiased? If not, find an unbiased estimator of and
denote it by ₁. [Hint: Derive the distribution of Y₁.]
(c) Show that ₂ = 2X, with X = X₁, is unbiased for 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb6b4902-24f8-4b5a-8abf-ecd52fa0b83e%2F8046b4b0-c72f-4f8b-9717-ed296d873c0f%2Fx3uvjs_processed.png&w=3840&q=75)
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- Let X1, , Xµ be iid with population density (1 0) I>0, Sx(x) = %3D otherwise. Here 0 is an unkown population parameter. 0 has an Exponential(1) distribution. Find the method of moment estimator for 0. Let's call this 6. Is ô unbiased for 0 ? Explain with precise computation. Show that X Find the maximum likelihood estimator for 0. Let's call this 62. Is ô2 unbiascd for 0 ? Explain with precise computation.Please answer number 3Let 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.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 -∞The differentiation approach to derive the maximum likelihood estimator (mle) is not appropriate in all the cases. Let X₁, X2,,X₁ be a random sample of size n from the population of X. Consider the probability function of X fe-(2-0), if 0Question 6 Suppose A and B are two events with probabilities: Find the following: a) P(ANB). b) P(A). c) P(Bº). d) P((An B)*). Edit View Insert Format Tools Table 12pt v P(Aª) = .40, P(B) = .25, P(A U B) = .75. ✓ Paragraph BIUAT² : B Q Search V7 Find the expected value of the function g (X) = X², where x is a random variable defined by the đensity, fx ) = a. eax, u u (x), where 'a' is a constant.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) = 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.Let x₁ = 6, x2 = 4, x3 = 5, x4 = 15 be a random sample from a population with probability density function a 10a (x + 10)α+1 Compute â, the maximum likelihood estimate of a. f(x) = You do not have to verify that your estimate is a maximum. 2 x > 0.Let {Xi}ni be IID from Uniform(θ,0). Find the likelihood function and the MLE of θLet X₁, X2, Xn be a random sample from a population with cumulative distribution function F(x) = xº, 0 0. ...Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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