I. The following numbers represent a random sample from a normal distribution with variance of 16 and unknown mean p:o

7.64 6.38 6.06 5.59 2.03 1.17 8.42 -6.83 2.25 5.78 7.56 4.33'

a.  What is the distribution of the maximum likelihood estimator (mle) for p?

b. Give the value of the mjg for g for this data. o

c.  Create a 90% confidence interval for p.o

2. If Z is a standard normal random variable, then Z² is a chi-squared random variable with 1 degree of freedom. The moment generating function of a chi-squared random variable with y degrees of freedom is mx(t) = (1 - 24¹². Suppose that Xi, X2, , Xn represent a random sample from a normal distribution with mean t and x - )² variance a². Use moment generating functions to find the distribution of ErL_i

3.  Let Xi, X2, , Xn be a random sample from a distribution with density fix(x) = Ox" for 0 < x <1 and 9 > O. Find the pit for 0.4j