1. Suppose X is a discrete random variable with the following pmf: 2 f 33 x=0 | 1 3'9 x=1 Pix): 2 . 1309) x22 [Zia9) x=3 where 0 5 9 5 1 . The following 10 independent observations were sampled from the distribution: (3,0,2,l,3,2,1,0,2,1) (a) (5 points) Find the method of moments estimate of 9. (b) (5 points) Find the maximum likelihood estimate of 9. (c) (5 points) Which estimate is better? Use math to support your answer. 2. A certain cylindrical machined part has nominal outside diameter of 25mm. Data for quality control measurements of 15 of these parts is provided in the le od.csv. (a) (5 points) Calculate the upper bound of an exact one-sided 95% condence interval for the variance of the diameter measurements. Explain why you believe this is an exact interval. (b) (5 points) Find the upper bound of the one-sided 95% condence interval using a bootstrap percentile interval. Compare the result to that from part (a) and provide a rationale for any observed differences. (6) (5 points) Test the research hypothesis that the standard deviation of the outside diameter measurement is less than 0.1mm. 3. (5 points) Given a distribution for the random variable XWith parameter 0;, E00: 1/3 , and Var(X)= 21(31): + 1) , what is the method of moments estimator for a