1) The following numbers represent a random sample from a normal distribution with variance of 16 and unknown mean µ:
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 µ? What is its distribution? Is the mle unbiased for µ?
Give the value of the mle for µ for this data.
b) Create a 90% confidence interval for µ.
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 γ degrees of freedom is mX(t) = (1 - 2t)-γ/2. Suppose that X1, X2, ... , Xn represent a random sample from a normal distribution with mean µ and variance σ2. Use moment generating functions to find the distribution of
3) Let X1, X2, ... , Xn be a random sample from a distribution with density fX(x) = θxθ-1 for 0 < x <1 and θ > 0. Find the mle for θ.