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