Statistics 100C Homework 1 Exercises

| August 30, 2017

Exercise 6 Consider the regression model yi = ?0 + ?1xi + i . Find cov(ei , ej ).

Exercise 7 Suppose Yi = ?xi + i . In this equation X is non-random, ? is a parameter (unknown), and ? N(0, ?). a. Find the mean of Y . b. Find the variance of Y . c. What distribution does Y follow? d. Write down the likelihood function based on n observations of Y and x. e. Find the maximum likelihood estimate of ?. Denote it with ?ˆ. f. Show that the estimate of part (e) is unbiased estimator of ?. g. Find the variance of this estimate.

Exercise 8 Consider the model of exercise 7. Find the covariance between ei and ej . Also find the variance of ei .

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