MGSC 372 Fall 2014
Assignment 1 Prof. B. Smith Due Oct 4,

QUESTION 1

The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a probability distribution that may be used to describe social, scientific, geophysical, actuarial, and many other types of observable phenomena.

Pareto originally used this distribution to describe the allocation of wealth among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society. He also used it to describe distribution of income.[8] This idea is sometimes expressed more simply as the Pareto principle or the "80-20 rule" which says that 20% of the population controls 80% of the wealth. This distribution is not limited to describing wealth or income, but to many situations in which an equilibrium is found in the distribution of the "small" to the "large". The following examples, among others, are sometimes seen as approximately Pareto-distributed: * The sizes of human settlements (few cities, many hamlets/villages. * File size distribution of Internet traffic which uses the TCP protocol (many smaller files, few larger ones) * Hard disk drive error rates
The Pareto distribution function may be given as follows: fxx0,θ= θx0θx-θ-1 , x ≥ x0, θ>1 a) Assume that x0>0 is given and that x1, x2, ….,xn is an i.i.d. sample. Find the MLE of θ.

b. We are told that x0=3. Given the following sample of five observations: x1=3, x2=2, x3=3, x4=6, x5=5. Find the MLE of θ for this sample. θ=1ln 3+ln2+ln3+ln6+ln55 –ln3=11.2583-1.0986=6.2617

c. Construct an Excel spreadsheet showing the values of the likelihood function