Consider a random sample X1,..., Xn with n> 3 from the exponential distribution with parameter > 0 and density f(x;A) = e-=/A, z>0. Consider the following two estimators of A: 3 X;, Ã =nX(1). 3 i=1
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- Let X₁, X2,..., X₂, be a random sample from a population with cumulative distribution function F(x) = xº, 0 0.Let X₁, ..., X be a random sample from a gamma distribution with a x = 2 and 3 = = 0. '1' 1. Find the simplified likelihood of 0 given x: L(01x). 2. Find the maximum likelihood estimator of 0,0 MLE* is unbiased. 3. Show that the value you found for 0 MLELet Y1 0 , zero elsewhere. Find Pr(Y1 < 1/3)
- For a random sample X1, X2, ..., X, following an Exponential(rate = 1/μ) distribution, we have seen that the large sample distribution of the sample mean X follows a Normal distribution with mean parameter µ and the variance parameter μ²/n. Suppose that we have a sample of size n = 25 from an exponential population with mean μ and we want to test n Hoμ = 2 v/s Hд μ # 2. Use the large sample distribution of ☑ to compute an approximate p-value for the test if we observe x = 2.6 and state the conclusion of your test at the 5% level of significance.k = Question 5 Suppose the random variables X and Y have a pdf given by f (x, y) = x+y on 0 Y). prob = b. Find F (,글): ans = Question 6 ch 30 HisenseLet X₁,..., Xn be a random sample from a distribution with one of two pdfs. If 0 = 1, then f(x;0=1) = 1 (0 < x < 1). If 0 = 2, then f(x;0= 2) = 2x1 (0 < x < 1). (a) Give a general form of the MLE of 0. (b) You are given a sample data of 3 values: 0.4, 0.5, 0.8. Find the MLE estimate of 0 based on the data.
- N= 150 observations were collected on a time series that was identified as a AR(2) time series. The following statistics were computed from the data. Mean - 45.0 Variance 15.6 Autocorrelation function (up to lag 5) r1 = 0.80, r2 = .50, rз = .26, r4 = -.10, rs = 0.08: == Estimate the parameters of the model using the method of moments.This exercise requires the use of a graphing calculator or computer programmed to do numerical integration. The normal distribution curve, which models the distributions of data in a wide range of applications, is given by the function 1 p(x) = e (x - 1)?/(202), V 2n 0 where n = 3.14159265... and o and u are constants called the standard deviation and the mean, respectively. Its graph (for o = 1 and u = 2) is shown in the figure. f(x) 0.3 0.2 0.1 3 4 5 With o = 3 and u = 0, approximate p(x) dx. (Round your answer to four decimal places.)Consider a set of data x1, x2, n n i=1 ..., n taken from a population with mean µ. - Show that (x-μ)² = Σ(x₂ − x)² + n(x − µ)². i=1
- Let X1, X2, ..., X50 be a random sample of size 50 from a distribution with density for 0Let Y < Y, < Y3 be the order statistic of a random sample of size 3 from the uniform distribution having pdff (x;0) = 1/0,0Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. FreemanMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. Freeman