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Chapter 4 - Random Variables
Todd Barr
22 Jan 2010
Geog 3000
Overview
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
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Discuss the types of Random Variables
Discrete
Continuous
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Discuss the Probability Density Function
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What can you do with Random Variables
Random Variables
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Is usually represented by an upper case X
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is a variable whose potential values are all
the possible numeric outcomes of an
experiment
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Two types that will be discussed in this
presentation are Discrete and Continuous
Discrete Random Variables
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Discrete Random Variables are whole numbers
(0,1,3,19.....1,000,006)
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They can be obtained by counting
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Normally, they are a finite number of values, but
can be infinite if you are willing to count that high.
Discrete Probability Distribution
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Discrete Probability Distributions are a description
of probabilistic problem where the values that are
observed are contained within predefined values
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As with all discrete numbers the predefined values
must be countable
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They must be mutually exclusive
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They must also be exhaustive
Discrete Probability Distro
Example
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The classic fair coin example is the best
way to demonstrate Discrete Probability
Classic Coin Example
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Experiment: Toss 2 Coins and Count the
Numbers of Tails
Physical Outcome
Value (x)
Probabilities, p(x)
Heads, Heads
0
¼ .25
Heads, Tails
Tails, Heads
1
½ .5
Tails, Tails
2
¼ .25
Classic Coin Continued
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Histogram of our tosses
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
Classic Coin Toss Summary
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Its easy to see from the Histogram on the previous
slide the area that each of the
results occupy
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If we repeat this test 1000 times, there is a strong
probability that our results will
resemble the previous Histogram but with some standard
variance
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For more on the classic coin toss and Discrete
Random Variables please go see the Educator video
at http://www.youtube.com/watch?v=T6eoHAjdAfM
Its all Greek to Me
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Mean and Variables of Random Variables,
symbology
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μ (Mu) is the symbol for population mean
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σ is the symbol for standard deviation
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s or x-bar are the symbols for data
Mean of Probability Distribution
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The Mean of Probability Distribution is a weighted
average of all the possible values within an
experiment
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It assists in controlling for outliers and its
important to determining Expected Value
Expected Value
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Within the discrete experiment, an expected value
is the probability weighted sums of all the potential
values
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Is symbolized by E[X]
Variance
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Variance is the expected value from the
Mean.
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The Standard Deviation is the Square Root
of the Variance
Continuous Random Variables
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Continuous Random Variables are defined by
ranges on a number line, between 0 and 1
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This leads to an infinite range of probabilities
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Each value is equally likely to occur within this
range
Continuous Random Variables
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Since it would nearly impossible to predict
the precise value of a CRV, you must
include it within a range.
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Such as, you know you are not going to get
precisely 2” of rain, but you could put a range at
Pr(1.99≤x≤2.10)
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This will give you a range of probability on the bell
curve, or the Probability Density Function
Probability Density Function
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The Probability Density of a Continuous Random
Variable is the area under the curve between points a
and n in your formula
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In the above Bell Curve the Probability Density
formula would be Pr(.57≤x≤.70)
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For a more detailed explanation please see the Khan
Academy at www.KhanAcademy.org
Adding Random Variables
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By knowing the Mean and Variance of of a
Random Variable you can use this to help predict
outcomes of other Random Variables
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Once you create a new Random Variable, you can
use the other Random Variables within your
experiment to develop a more robust test
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As long as the Random Variables are independent
this process is simple
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If the Random Variables are Dependent, then this
process becomes more difficult
Useful Links
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PatrickJMT
http://www.youtube.com/user/patrickJMT
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The Khan Academy
http://www.KhanAcademy.org
➲ Wikipedia
http://www.wikipeda.org