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- I. The length of time required by students to complete a 1-hour exam is a random variable X (in hours) with probability density function f(x) = 45(x3 + 1) for 0 < x < 1 and f(x) = 0 for x • • • • • • otherwise. Randomly select a student from this class. Whatistheprobabilitythatthecompletiontimeforthisstudentisatleast50minutes? Note that 40 minutes is equal to 40/60 hr. What is the probability that completion time is between 20 and 40 minutes? Determine the mean and variance of completion time. What is the CDF Graph the CDF Graph the PDF II . The average American eats 2,000 calories per day with a standard deviation of 500 calories. If this distribution follows a normal distribution, (a) What is the probability that someone eats between 2,000 and 3,000 calories per day? (b) How many calories corresponds to the 70th percentile? III. Draw a picture (Venn Diagram) for the following distributions. Additionally, find the fol- lowing probabilitis if P(A) = 0.4 and P(B) = 0.3 and P(A ∩…b) Let X to be a random sample from a distribution with a probability density function given by f(x; 0) = {0x0-¹,if 0 i) Find the level of significance of the test, a. ii) Calculate the power of the test.Suppose the joint distribution of the two random variables X1 and X2is given by S6x2, 0< x2 < x1 < 1, (o, f(x1, 22) = - elsewhere. (a) Find the marginal distribution fx, (x1) of X1 and verify that it is a valid density function. (b) What is the probability that X, is less than 0.5, given that X, is 0.7?
- 1. Let X1,..., X, be a random sample with each X, having probability density function f(r;0) = 36°x¬*, 0< o²The 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.c) Let Y₁, Y₂,..., Yn be a random sample whose probability density function is given by f(v:B)= 684 - fa 00 0, elsewhere 200 200 200 and suppose that n = 200, y = 20, y = 100, y = 250 and $ = 0.025. i=1 i) Derive the standard error of ß, se(B) = 0.0009, using MLE approach. ii) Find an approximate 95% Confidence interval for B.9. The random variable X has probability density function, f(x) where k 12B. (a) An exponential distribution has probability density function ƒ(x) = ¹²e-x/a, x ≥0. α Show that the mean of the distribution is a and the variance is a². (b) A machine contains two belt drives, of different lengths. These have times to failure which are exponentially distributed, with mean times a and 2a. The machine will stop if either belt fails, and the failures of the belts are independent. Show that the chance that the machine is still operating after a time a from the start is e 32b) Let Y,,Y2, .. , Yn denote a random sample from N(0,0) distribution with probability density function: f(y;8) = e V2n0 i) Show that f(y; 0) belongs to the 1-parameter exponential family. ii) What is the complete sufficient statistic for 0? Justify your answer. iii) Show whether or not, the maximum likelihood estimator is an unbiased estimator of 0. iv) Does the estimator attains the minimum variance unbiased estimator of 0.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