a) Suppose that X₁, X₂, X3....X₁ is a random sample of n from population X which has a chi- square distribution with one degrees of freedom. (i) Find lim M, (t). 71-00 (ii) Use the Central Limit Theorem to compute P(0.5 < X < 1.5), for n = 30. (iii) Use the Chebychev inequality to find the value of k for P(0.5 < X < 1.5), if n = 30.
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Suppose that X₁, X2, X3.....X₁ is a random sample of n from population X which has a chi-
square distribution with one degrees of freedom.
(i) Find lim Mg(t).
71-00
(ii) Use the Central Limit Theorem to compute P(0.5 < X < 1.5), for n = 30.
(iii) Use the Chebychev inequality to find the value of k for P(0.5 < X < 1.5), if n = 30.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdffa0d4f-9671-41bf-b34d-8e971e176b9f%2F8244becd-d9d7-4966-8fd5-0d670132a76b%2F7eeq3h_processed.jpeg&w=3840&q=75)
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- (i) Let X ~ Poisson(A). Then, using Chebyshev's inequality, show that P(X > 21) < 1/A. (ii) Suppose that the number of errors per computer program has a Pois- son distribution with mean 5. We get 125 programs. Let X1, X2, ..., X125 be the number of errors in the programs. Then, using the central limit theorem, find an approximate value for P(Xn < 5.5).6Suppose that the claim size distribution of an insurance portfolio follows a Pareto distribution of the form α+1 α f(x) β = B\B+x (i) Derive a formula for the rth moment, ar, of this Pareto distribution in terms of its (r-1) th moment, αr-1. Show your steps clearly with reasons. (ii) From this expression find α3 and α4 using the known result for the mean μ of the above Pareto distribution. (You may assume that α>r, where a is one of the two parameters of the Pareto distribution).
- 11 Assume the GPAS of high school students, X, follows a continuous uniform on the domain 12.0.3 Problem 3: Central Limit Theorem Here we will verify the Central Limit Theorem and reproduce plots similar to those from Wikipedia (https://en.wikipedia.org/wiki/Central_limit_theorem#/media/File:Dice_sum_central_limit_theorer a) Write a function that returns n integer random numbers, uniformly distributed between 1 and 6, inclusively. This represents n throws of a fair 6-sided die. The value that comes up at each throw will be called the "score". b) Generate a distribution of 1000 dice throws and plot it as a histogram normalized to unit area. Compute the mean ₁ and standard deviation o₁ of this distribution. Compare your numerical result to the analytical calculation. c) Generate 1000 sets of throws of N = 2, 3, 4, 5, 10, 20, 30 dice, computing the total sum of dice scores for each set. For each value of N, plot the distribution of total scores, and compute the mean and standard deviation on of each distribution. This should be similar to the plot at the link above. d) Plot the…b) Let X₁, X₂,..., X and Y₁, Y₂, ..., Ym be random samples from populations with moment generating 25 functions Mx₁(t) = ³t+t² and My(t) = (₁-¹)²5, respectively. i) Find the sampling distribution of the statistic W = X₁ + 2X₂ − X3 + X4 + X5. ii) What is the value of the sample size n, if P[Σ1(X¡ − X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Ỹ - µy| ≥10) < 0.04?5. Does f(x) = In(x) satisfy the hypotheses of the Mean Value Theorem on [1,4]? If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem.20. Let X1, X2, ables, and set X, be independent, Exp(a)-distributed random vari- Y1 = X(1) and Y = X(k) – X(k-1); for 2Given that Z - N(O, 1) use tables to evaluate correct to three decimal places then P(1.24 < Z < 2.36)=Solve all plzLet X₁,..., Xn be a random sample from a geometric distribution, X~ GEO(p). Here, -1 P[X = x] = p(1 − p)*−¹ for x = The method of moments unbiased estimator for Var [X] None of the other answers ·Σ" (X; – X)² i= X² n 1 [X² -x] = 1, 2,... n+1 = 1-p p² is7. (12) Show that the hypotheses of the Mean Value theorem are satisfied and find all values satisfying the conclusion for 3 - 7? if r 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