Chapter 4 - Random Variables

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## Chapter 4 - Random Variables

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**Todd Barr**22 Jan 2010 Geog 3000 Chapter 4 - Random Variables**Overview**• Discuss the types of Random Variables • Discrete • Continuous • Discuss the Probability Density Function • What can you do with Random Variables**Random Variables**• Is usually represented by an upper case X • is a variable whose potential values are all the possible numeric outcomes of an experiment • Two types that will be discussed in this presentation are Discrete and Continuous**Discrete Random Variables**• Discrete Random Variables are whole numbers (0,1,3,19.....1,000,006) • They can be obtained by counting • Normally, they are a finite number of values, but can be infinite if you are willing to count that high.**Discrete Probability Distribution**• Discrete Probability Distributions are a description of probabilistic problem where the values that are observed are contained within predefined values • As with all discrete numbers the predefined values must be countable • They must be mutually exclusive • They must also be exhaustive**Discrete Probability Distro Example**• The classic fair coin example is the best way to demonstrate Discrete Probability**Classic Coin Example**• Experiment: Toss 2 Coins and Count the Numbers of Tails**Classic Coin Continued**• Histogram of our tosses**Classic Coin Toss Summary**• Its easy to see from the Histogram on the previous slide the area that each of the results occupy • 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 • 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**• Mean and Variables of Random Variables, symbology • μ (Mu) is the symbol for population mean • σ is the symbol for standard deviation • s or x-bar are the symbols for data**Mean of Probability Distribution**• The Mean of Probability Distribution is a weighted average of all the possible values within an experiment • It assists in controlling for outliers and its important to determining Expected Value**Expected Value**• Within the discrete experiment, an expected value is the probability weighted sums of all the potential values • Is symbolized by E[X]**Variance**• Variance is the expected value from the Mean. • The Standard Deviation is the Square Root of the Variance**Continuous Random Variables**• Continuous Random Variables are defined by ranges on a number line, between 0 and 1 • This leads to an infinite range of probabilities • Each value is equally likely to occur within this range**Continuous Random Variables**• Since it would nearly impossible to predict the precise value of a CRV, you must include it within a range. • 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) • This will give you a range of probability on the bell curve, or the Probability Density Function**Probability Density Function**• The Probability Density of a Continuous Random Variable is the area under the curve between points a and n in your formula • In the above Bell Curve the Probability Density formula would be Pr(.57≤x≤.70) • For a more detailed explanation please see the Khan Academy at www.KhanAcademy.org**Adding Random Variables**• By knowing the Mean and Variance of of a Random Variable you can use this to help predict outcomes of other Random Variables • Once you create a new Random Variable, you can use the other Random Variables within your experiment to develop a more robust test • As long as the Random Variables are independent this process is simple • If the Random Variables are Dependent, then this process becomes more difficult**Useful Links**• PatrickJMT http://www.youtube.com/user/patrickJMT • The Khan Academy http://www.KhanAcademy.org • Wikipedia http://www.wikipeda.org