8.3.12. Let Y₁Y₂ << Ys be the order statistics of a random sample of size n = 5 from a distribution with pdf f(x; 0) = e--, -∞0
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- 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 MLE= 16. A simple random sample of size n = 64 is obtained from a population with u = 77 and o = (a) Describe the sampling distribution of x. (b) What is P (x>79.5) ? (c) What is P (xs72.9)? (d) What is P (75.9 79.5) = (Round to four decimal places as needed.) (c) P (xs72.9) = (Round to four decimal places as needed.) (d) P (75.9Let X. X , be a random sample of siren from a normal distribution with population mean 0 and population variance a>Ois unknown, that is N(0, ). Suppose that we define the following plvotal quantity (n-1)s where & and S' are the sample mean and the sample variance, respectively. Use the plvotal quantity (Q) to derive the (1- a) 100% confidence interval (C) for the unknown parameter (o) if the two critical values are q--n, a) and (n-1) (A) CI = (n8+(n-1)51 + n-1) s (8) CI - (C) CI - + (n-1) s (D) CI= O D O A O c O BSuppose the random variable, X, follows a geometric distribution with parameter (0< 0 <1). Let X₁, X₂, X, be a random sample of size n from the population of X. (d) Find the method of moments estimator (mme) of 0. (e) Show how the mle of 0 is related to its mme. Explain if it is always true. (f) Verify if X, is a sufficient statistic for 8 or not. (g) Justify if = is an unbiased estimator for 6.Let X1, X2, *. Xn be a random sample of size n from norm population with mean and unknown a. Let X and s be the sample mean and sample standard deviation. Then for any 0Q2: Let x1,X2, . , Xn and y1, y2, ..., Ym represent two independent random samples from the respective normal distributions N(H1,07) and N (H2, 03). It is given that of = 30, but ožis unknown. Then 1. A random variable that can be used to find a 95% confidence interval for - is (x- y) - (H1 - H2) А. ns3 + ms n+m. 3(n+m-2)"tm, nm O A (x – y) – (41 – H2) В. ns? + ms; n + m. (n + m – - 2) пт (x – y) – (41 - H2) С. ns3 + mS;_n+3 3(n+ m-2) ("+3m nm ОсThis column contains data x1, . . . , xn, which is assumed to be a sample from a normal distribution N(µ, σ 2 ). For this sample, find the maximum likelihood estimates µˆ and σˆ.3) Explain what is meant by constructing a confidence interval for an unknown parameter 0 from a given sample x, ..., N. Let a family of PDFS fS(x: 0). -o 0, S(x: 0) = 0. x 0. Explain how to construct a 95% confidence interval for a from a sample x, . , x. Justify the claims about the distributions you use in your construction.18. Show that for type lII population - x/0 dp OQ5 Find the variance for the PDF px(x) = e-«/2, x > 0.Let X1, X2, ..., X, be a random sample from f(x;0) = 0e-tI(0,00)(x). Find a 100y or (1- a)100 percent confidence interval for the mean of the population.3.4 Solve the below problem: Let Y, and Y, have a bivariate normal distribution: 2. 1. 2. Y2-H2 exp 2no,02y1-p2 2(1-p2) {. (ज-र.) + ( ज-र) (ज-पदर) 07 (जब)} Dr= –00 < y, < ∞, -00 < y2 < ∞, Show that the marginal distribution of Y, is normal with mean µz and variance o is given by: 2, 1 f(y2) = 203 ,-00 < Y2 < ∞ 2.SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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