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- 18. Show that for type lII population - x/0 dp OQuestion 3. Let X₁,, Xn be a random sample from a distribution with the pdf given by f₁₁x(x) = exp(-ª) if x ≥ A, otherwise f(x) = 0, where > 0. Find the MLE's of 0 and X. Start by writing the likelihood function and note the constraint involving A. Question 4. #4.2.9 of the textbook.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 0Suppose that X₁,..., Xn is a random sample from a distribution with probability den- sity function 2 ) = { +/- - - fx(x) = √3/₂e-x/0, 0 0. You are provided the information that the maximum likelihood estimator of is Ô i=1 = 2n (You do not need to derive this mle.) (a) Verify that Fisher's information in the random sample is given by 2n In (0) 027. For simple linear regression, we assume that Y = Bo + BIX +e, where e - N(0,0?) and X is fixed (not random). We collect n i.i.d, training sample (x),y1)....(XYn)). Prove that the (Bo.B1) estimated through minimizing RSS equals to the one through maximizing likelihood.Let X1,..., X, a random sample with distribution f(x; 0) = (0 + 1)æº, 0 -1. (a) Find the method of moments estimator of 0. (b) Show that the method of moments estimator is consistent. (c) Find the maximum likelihood estimator of 0. (d) takes value 0 = 1.2. Use the parametric bootstrap method to obtain a 95% revised bootstrap percentile confidence interval using the maximum likelihood estimate. Make sure to include a plot of the bootstrapped values and an interpretation of your confidence interval. Given a sample of size n = 500, suppose that the maximum likelihood estimate of 0If the roots of the quadratic equation x – ax + b = 0 are real and b is positive but otherwise unknown, what are the expected values of the roots of the equation. Assume that b has a uniform distribution in the permissible range. -B we regression Given the data (Xi) and (1₁) will assume that a model Y₁ =B₁Xi + Ei is with normally distributed independent Error and í 12 3 4 Xi 7 12 25 30 Yi 14 17 30 42 a) State the likelihood function for the four y observations. b) Evaluate the likelihood function for By and Bo and B₁ = 1 and B₁ = 2- For which of these is the likelihood • function the largest.2) Let X₁, X2, ..., Xn be a random sample from the pdf f(x;0) = 0xª−¹, 0≤x≤ 1,0 <0 < ∞. Find the MLE of 0 and show that its variance approaches 0 as n approaches ∞o.Suppose that X₁ = 1; X₂ = 1; X3 = 0; X₁ = 1; X5 = 1, X6 = 1; X₂ = 0; X = 1; X9 = 0, X10 = 0, represents a random sample. Each of these X's comes from the same population and has a density of fx₁ = 0¹-* (1 - 0)*; x = 0,1 First determine the form of the maximum likelihood estimator for 0, then use the MLE formula and the data provided to find an estimate for 0. Round your answer to 3 decimal places. (Answer is 0.4)PLease Show all work so I can UnderstandLet {Xi}ni be IID from Uniform(θ,0). Find the likelihood function and the MLE of θRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON