Homework 3, Statistics 512, Spring 2005

Homework 6, Statistics 512, Spring 2005

This homework is due Thursday, March 3rd at the beginning of class.

1. Hogg, McKean and Craig, 6.1.5. e-l l k 2. Suppose X1, , X n are iid Poisson(l ): P( X= k ) = i k! (a) Find the maximum likelihood estimate of l . (b) Find the Fisher information I(l ) . What is the asymptotic distribution of ˆ n(lMLE - l ) ? (c) The Poisson distribution has been used by traffic engineers as a model for light traffic, based on the rationale that if the rate is approximately constant and the traffic is light (so the individual cars move independently of each other), the distribution of the counts of cars in a given time interval or space area should be nearly Poisson. The following table shows the number of right turns during 300 3-minute intervals at a specific intersection. Find the maximum likelihood estimate of l for these data and an approximate 95% confidence interval for l . Number of right Frequency Number of right Frequency turns turns 0 14 7 14 1 30 8 10 2 36 9 6 3 68 10 4 4 43 11 1 5 43 12 1 6 30 13+ 0

3. Hogg, McKean and Craig, 6.1.12.

4. Suppose that in a population of twins, males and females are equally likely to occur and that the probability that twins are identical is a . If twins are not identical, their genes are independent.

(a) Show that 1+a P(both twins are male) = P(both twins are female) = 4 1-a P(one twin is male, one twin is female) = 4

(b) Suppose that 1000 twins are sampled. It is found that 400 twins are both male, 430 twins are both female and for 170 twins, one is male and one is female. Find the maximum likelihood estimate of a .

(c) Find an approximate 95% confidence interval for a .