Discrete Probability Distributions Last Update 5th May 2011 SESSION 35 & 36 Discrete Probability Distributions

Lecturer: Florian Boehlandt University: University of Stellenbosch Business School Domain: http://www.hedge-fund-analysis.net/pages/vega.php

Learning Objectives All measures for grouped data: Poisson Experiment Poisson Probability Distribution

Binomial and Poisson distribution Quantitative Data Data Data variables can only assume certain values and are collected typically by counting observations Data variables can assume any value within a range (real numbers) and are collected by measurement. Discrete Continuous Continuous probability distribution functions (most notably normal, student-t, F, Chi-Squared) Binomial and Poisson distribution

What is a Poisson Experiment? A Poisson experiment, like the binomial variable, is a discrete probability distribution. In contrast to the binomial random variable which defines the number of successes in a set number of trials, the Poisson random variable is the number of successes in an interval of time or specific region of space. Examples: The number of cars arriving at a service station in one hour (time interval: 1 hour.) The number of flaws in a bolt of cloth (specific region: bolt of cloth) The number of accidents in 1 day on a particular stretch og highway (both time interval and specific region)

Binomial versus Poisson In the event where no specific reference is made to the number of trials (such as in games of chance like blackjack, poker, lottery), and provided that the other properties of the Poisson distribution hold, a Poisson experiment will contain reference to a time frame, or alternatively, a region of space. For example: Per hour, per day, per week, within a year, during an average month etc. Per square feet, per 100 meters etc. This is only intended as a rough reference point and does not serve as a precise definition!

Definition Poisson Experiment A Poisson experiment is characterized by the following properties: The number of successes in any interval is independent of the number of successes that occur in any other interval The probability of success in an interval is the same for all equal-size intervals The probability of success in an interval is propoertional to the size of the interval The probability of more than one success in an interval approaches zero as the interval becomes smaller.

Poisson Random Variable A Poisson random variable is the number of successes that occur in a period of time or an interval of space in a Poisson experiment. The Poisson probability distribution is defined as: Where μ is the number of successes in the interval or region and e (≈ 2.71828) is the base of the natural logarithm.

Example Probability of the number of typographical errors in textbooks: A statistics instructor has observed that the number of typographical errors in new editions of textbooks varies considerably from book to book. After some analysis, he concludes that the number of errors is Poisson distributed with a mean of 1.5 per 100 pages. The instructor randomly selects 100 pages of a new book. What is the probability that there are no typographical errors?

Solution Variable Value Interval 100 Number of errors per 100 1.5 μ (Number of errors / interval) e Euler (see calculator e) x The probability that in the 100 pages selected there are no typographical errors is P(0) = 0.2331.

Exercises Probability of the number of typographical errors in textbooks: The newly received statistics book now contains 400 pages. Everything else being as before: What is the probability that there are no typos? What is the probability that there are five or fewer typos? Comment: Note that for problems of type b) it may be easier to use pre-computed probabilities for a large number of problems from tables in textbooks or statistical processing software (including Excel). The same is true for the binomial distribution.