Let X1, X2, ...Xn be a random sample from a Uniform(0, 10) distribution. (a) Show that...

 

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Let X1, X2, ...Xn be a random sample from a Uniform(0, 10) distribution. (a) Show that the MLE of 0 is, 0ML = min{X1, X2, ...X„}. (b) Find the sampling distribution of the MLE by first finding the CDF and then the PDF (by differentiating the CDF). (c) Find the mean and variance of the MLE. Show that the MLE is biased. What happens to the bias as n gets larger and larger? (d) Say we observe a sample: 6.44392824 4.14275897 0.37427695 5.90141461 1.05512205 1.36180572 1.20312096 3.66009730 9.22633586 2.65242304 2.53760655 1.02514405 2.42837809 0.04037553 3.87637621 2.47093808 2.35322481 9.45535216 7.25094234 ..52191867 Find the maximum likelihood estimate of 0 given this dataset. Also use this to estimate the mean and variance of the maximum likelihood estimator. Let X1, X2, ...Xn be a random sample from a Uniform(0, 10) distribution. (a) Show that the MLE of 0 is, 0ML = min{X1, X2, ...X„}. (b) Find the sampling distribution of the MLE by first finding the CDF and then the PDF (by differentiating the CDF). (c) Find the mean and variance of the MLE. Show that the MLE is biased. What happens to the bias as n gets larger and larger? (d) Say we observe a sample: 6.44392824 4.14275897 0.37427695 5.90141461 1.05512205 1.36180572 1.20312096 3.66009730 9.22633586 2.65242304 2.53760655 1.02514405 2.42837809 0.04037553 3.87637621 2.47093808 2.35322481 9.45535216 7.25094234 ..52191867 Find the maximum likelihood estimate of 0 given this dataset. Also use this to estimate the mean and variance of the maximum likelihood estimator.

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