Prove Theorem Let X be a random variable with the normal distribution S(x) = z ²] exp[=1/2(x– µ)²lo Then (i) ECX) =µ and (ii) var(X) = o. Hence Ox = 0. %3D
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- 43. The probability density functions of two statistically independent random variables X and Y are fx(x) = }u(x - 1)e-lx-1)/2 fr(y) =u(y- 3)e-(s-3)/4 %3D Find the probability density of the sum W =X+Y.E [X] = 4 and In X follows a normal distribution with a variance of 3. Express the value of E [min (X - 4, 1)] as the cumulative distribution function for the standard normal distribution.
- EL 466 416 13.) The continuous random variable (RV) X is uniform over [0,1). Given Y = -ln X what is P({07. Let X~ N (0,0²) and {X; : i = 1,2,..., n} be a random sample from X. (a) Formulate the log-likelihood function. (b) Find the ML estimator of o². (c) Derive the variance of the ML estimator of o2, 62. Does the variance of ô2 achieve the CR bound? (d) Derive the asymptotic distribution of √n (-o).9. The random variable X has probability density function, f(x) where k 1Q3/ consider a random variable X with density function 0.125(4 + 2x) 1< x < 2 f (x) = elsewhere Find the variance for g(X) = 4X +3Let Y be a random variable with the mgfM(t) = .35e^(−4t) + .1 + .25e^(2t) + .3e^(4t).(a) What is the pmf of Y?(b) Calculate the variance of Y directly from the pmf you determined in part (a).(c) Calculate E[Y^4] directly from the mgf.Recommended 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