Transcript Lesson 11 - hedge fund analysis

Last Update
5th May 2011
SESSION 35 & 36
Discrete Probability Distributions
Lecturer:
University:
Domain:
Florian Boehlandt
University of Stellenbosch Business School
http://www.hedge-fundanalysis.net/pages/vega.php
Learning Objectives
All measures for grouped data:
1. Poisson Experiment
2. Poisson Probability Distribution
Quantitative Data
Data variables
can only assume
certain values
and are collected
typically by
counting
observations
Data
Discrete
Continuous
Binomial and
Poisson
distribution
Continuous
probability
distribution
functions (most
notably normal,
student-t, F,
Chi-Squared)
Data variables
can assume any
value within a
range (real
numbers) and
are collected by
measurement.
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:
1. The number of successes in any interval is independent of
the number of successes that occur in any other interval
2. The probability of success in an interval is the same for all
equal-size intervals
3. The probability of success in an interval is propoertional to
the size of the interval
4. 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)
1.5
e
Euler (see calculator e)
x
0
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:
a) What is the probability that there are no typos?
b) 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.