Transcript Recommendation of a Strategy
Chapter 4 - Random Variables
Todd Barr 22 Jan 2010 Geog 3000
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 Physical Outcome Heads, Heads Heads, Tails Tails, Heads Tails, Tails Value (x) 0 1 2 Probabilities, p(x) ¼ .25
½ .5
¼ .25
Classic Coin Continued
➲ Histogram of our tosses 0.6
0.5
0.4
0.3
0.2
0.1
0 0 1 2
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.Khan
Academy.org
➲ Wikipedia http://www.wikipeda.org