1. Find the method of moments estimator for the parameter p of a geometric distribution.
2. Let X be normal with mean p. and variance a2 both of which are unknown. Find the method of moments estimators for these parameters. Are the estimators obtained unbiased for their respective parameters? Explain.
3. Carbon dioxide is an odorless, colorless gas which constitutes about .035% by volume of the atmosphere. It affects the heat balance by acting as a one-way screen. It lets in the sun heat to warm the oceans and the land but blocks some of the infrared heat that is radiated from the earth. This reflected heat is absorbed into the lower atmosphere producing a greenhouse effect which causes the earths surface to become warmer than it would be otherwise. Systematic measurements of CO2 began in 1957 with Charles D. Keeling began monitoring at Mauna Loa in Hawaii. (McGraw-Hill Yearbook of Science and Technology, 1983, p. 68.)
(a) Assume that these CO2 readings (ppm) are obtained:
Construct a stem-and-leaf diagram for these data using 31, 32, 32,33,33,34,34.1: as stems, Graph leaves 0-4 on the first of each repeated stem and leaves 5-9 on: the other. Is it reasonable to assume that the CO2 level in the atmosphere is normally distributed? Explain.
(b) Estimate p, and a2 using the method of moments estimators.
(c) Find an unbiased estimate for a2.