*8. Let Y1,...,Yn iid Bernoulli(0). (a) Use the asymptotic distribution of the MLE to show that: Vn(Y – 0) d, N (0,1) VY(1 – Y) (b) Derive an approximate 1 - a confidence interval for 0 from your answer to the previous part.
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![*8. Let Y1, ...,Yn
iid
Bernoulli(0).
(a) Use the asymptotic distribution of the MLE to show that:
Vn(Y - 0) , N (0,1)
VÝ(1 – Y)
(b) Derive an approximate 1 – a confidence interval for 0 from your answer to the previous
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- Consider randomly selecting n segments of pipe and determining the corrosion loss (mm) in the wall thickness for each one. Denote these corrosion losses by Y₁' Yn. The article "A Probabilistic Model for a Gas Explosion Due to Leakages in the Grey Cast Iron Gas Mains"+ proposes a linear corrosion model: Y; = t;R, where t; is the age of the pipe and R, the corrosion rate, is exponentially distributed with parameter 1. Obtain the maximum likelihood estimator of the exponential parameter (the resulting mle appears in the cited article). [Hint: If c> 0 and X has an exponential distribution, so does cX.] O Â = O Â = O λ = * * O n Srit) i = 1 n j = 1 Σ i = 1 n n i = 1 n LY i = 1 n 0 1 = L (rt) 2 i = 1 | = 1 n n Y; | = 1Suppose X1,...,Xn are iid with PDF f(x | θ) = 2x/θ2, 0 < x < θ. (a) Write out the joint PDF f(x | θ), and write the likelihood function L(θ | x). (b) Find the MLE for θ. (c) Find the MLE for the median of the distribution.= 7.1.7. Let X1, X2,..., Xn denote a random sample from a distribution that is N(μ,0), 0 < 0 <∞o, where u is unknown. Let Y E(Xi-X)2/n and let L[0,8(y)] = [0-8(y)]². If we consider decision functions of the form 8(y) = by, where b does not depend upon y, show that R(0, 8) = (0²/n²) [(n² - 1)6² -2n(n-1)b+n²]. Show that b = n/(n+1) yields a minimum risk decision function of this form. Note that nY/(n+1) is not an unbiased estimator of 0. With 8(y) ny/(n+1) and 0<0<∞, determine max, R(0,8) if it exists. =
- Let W₁ < W₂ < ... < Wn be the order statistics of n independent observations from a U(0, 1) distribution. (a) Find the pdf of W₁ and that of Wn. (b) Use the results of (a) to verify that E (W₁) = 1/(n+1) and E(Wn) = n/(n+1).Let X1, X2, ..., Xn be a random sample from a normal distribution with mean u and variance o?. Find an unbiased estimator for o? and show that ΣΧ-Χ- E(X; - X)² = E(X²) – nX². i=1 i=1Q7. Let X; (i = 1,.,n) be a random sample from the N(u, o²) population with unknown parameters u and o?. Then we know that T, = E-1(X - X)²/(n – 1) is an unbiased estimator -xi-1 . Now consider two other estimators T, and T3 for o? where of a? and further, (n-1)T T2 = E(X – X)*/n and T3 = E(X – X)²/n + 1). %3D What is the variance of T,? Find out the variances and biases of T2 and T3 and thus obtain their m.s.e.'s. Show that among the three estimators T,, T2 and T3 the one with minimum m.s.e. is T3.
- 5.5.5 X and Yare random variables with the joint PDF S5x²/2 fx,x (x, y) - otherwise. (a) What is the marginal PDF fx(x)? (b) What is the marginal PDF fy(y)?(c) What is the asymptotic distribution of √n(0-0)? 6.2.9. If X1, X2,..., Xn is a random sample from a distribution with pdf ={ f(x; 0) = 303 (x+0)4 0 07. a) Suppose that X is a uniform continuous random variable where 0You test some components with a Weibull distribution at temperatures of 100 Celcius and 60 Celcius. At those temperatures you estimate a characteristic life of 15000 hours and 25000 hours respectively. hint: Boltzmann's constant is k = 8.617 x 10-5. Temperature in Celcius + 273.15 = temperature in Kelvin. If n is an integer, T(n) = (n – 1) a) Using linear regression, estimate the Arrhenius model parameters A and AH. b) Give the MTTF, median, and 90th percentile at T = 20 Celcius. Assume that the shape parameter is 0.57. 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.|20 points] Suppose Y1,. where 0 < 0 < 1 is an unknown parameter. That is, Yn is a random sample of size n from Uniform(0, 20], 1 f(y) if 0SEE MORE QUESTIONSRecommended 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