Let X be one observation from a population with pmf |x| f(x|0) = (9) ¹ (1 – 0)¹−|™|, x = −1, 0, 1; 0 ≤ 0 ≤ 1. 2 That is, X takes on values -1, 0, 1 with probabilities f(−1; 0) = ƒ(1; 0) = 0/2 and ƒ(0; 0) = (1 — 0). (a) Find the MLE of 0. =
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- 2. Suppose X1,X2,...,Xn is a random sample from a binomial distribution. Recall X is the numberof successes in m independent trials with probability of success at each trial is θ. We know thatE[X] = mθ and V ar(X) = mθ(1 −θ).(a) Find the method of moment estimator for θ.(b) Is the estimator in (a) unbiased?(c) Is the estimator in (a) consistent?(d) Show that the estimator in (a) is also the maximum likelihood estimator.3.2.13 Y is a continuous random variable with { 2(1- y), 0< y < 1; fy(4) = { 0, otherwise. Derive Fy(t).5. Suppose that X1, X2, .. , Xn are an iid sample from the distribution given below, f(X)e = (1/2)(1+ xe); –1 < X < 1. Show that 3X is a consistent estimator of e
- 34. Consider the continuous random variable X whose pdf is given by f(r) = (a-r³)Io<<1(1). (a) Find the value of a that makes f(z) a pdf. (b) Find E(X). (c) Find V(X). (d) Find P(X ≤).Q. 25 If a Poisson variate X is such that P (x = 1) = P(x=2), Work out P (x = 4).1. Suppose we have a random sample X₁, X2, ..., Xn from an Exponential distribution with mean 1/A. Suppose we want to estimate the mean 1/A. One estimator for 1/X is T₁ = X. Of interest is to note that the minimum of X₁, X2,..., Xn, say X(1), has an Exponential distribution with mean (n)-¹. (a) Show that T₁ is an unbiased estimator of 1/A. (b) Find the constant a such that T₂ = aX(1) is an unbiased estimator of 1/X. (c) Since T₁ and T₂ are both unbiased we prefer the estimator with smaller variance. Which of the estimators T₁ and T₂ would you choose for estimating the mean 1/X?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=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