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- Prove the propertyb) Let X₁, X₂,..., X, denote a random sample of size n from a distribution with probability density function: i) ii) iii) f(x; 0) = {0(1-x)(¹+0), Find the maximum likelihood estimator of 0. Derive fisher's information of 0. suppose that 5 independent observations were taken from such a distribution: 4 1 3 5 4 and 0 = 0.7052 Find a 99% confidence interval for 0. x>0 elsewhere DLet 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).
- 3 Let X be the number of siblings of WSU students. The pdf of X is: f(x) = (5-x)/15 for x = 0, 1, 2, 3, 4 What is the probability that a randomly selected student has 3 or more siblings if it is known that they are not an only child X 0.133 X 0.2 / 0.3 O X 0.667 X 0.933b) Let X₁, X2, X3,...,Xn be a random sample of n from population X distributed with the following probability density function: f(x;0)=√√2n0 0, -20₁ if -∞0 < x <∞0 otherwise (i) Find the parameter space of 0. (ii) Find the maximum likelihood estimator of 0. (iii) Check whether or not the estimator obtained in (ii) is unbiased. (iv) Find the Fisher information in this sample of size n about the parameter 0.The probability density function for the continuous random variable X is given by: (А(х? — 2х + 21) 0Let X be a point randomly selected from the unit interval [0, 1]. Consider the random variable Y = (1-X)-¹/2 (a) Sketch Y as a function of X. (b) Find and plot the cdf of Y. (c) Derive the pdf of Y. (d) Compute the following probabilities: P(Y > 1), P(3 < Y < 6), P(Y ≤ 10).b) The probability density function of random variable X is given as: - Вх {*5 if 0 < x < 1 otherwise f(x) = If the expected value of the random variable X is 1/3. Find the values A and B.The random variable X has expected value E(X)=4 and Var(X)=4. Here, we want to approximate the probability that X lies in the interval [3.4,5.4] using a transformation to a standard normal distribution Z. Then we will need to calculate the probability that Z lies in the interval [z1,z2] where z1= z2= Suggest z1 and z2 Do NOT attempt to make any sort of continuity correction.6. Suppose that the random variables X and Y have joint probability density function given by x+y, 0) Let Yhave a binomial distribution with parameters n and p. H,: p= 1 is rejected 1 is accepted if Y2 c. Find n and c to give a power function T(p) 2 and H,:p> so that a = 0.10 and T = 0.95 , approximately.Let Y1, Y2,., Ya be a collection of independent random variables with distribution function y 8 Show that Y converges in probability to a constant, and provide that constant. 1Recommended 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