probability distributions: binomial & normal ginger holmes rowell, phd msp workshop june 2006
TRANSCRIPT
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Probability Distributions: Binomial & Normal
Ginger Holmes Rowell, PhD
MSP Workshop
June 2006
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Overview
Some Important Concepts/Definitions Associated with Probability Distributions
Discrete Distribution Example: Binomial Distribution More practice with counting and complex
probabilities Continuous Distribution Example:
Normal Distribution
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Start with an Example
Flip two fair coins twice List the sample space:
Define X to be the number of Tails showing in two flips.
List the possible values of X Find the probabilities of each value of X
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Use the Table as a Guide
x Probability of getting “x”
0
1
2
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X = number of tails in 2 tosses
x Probability of getting “x”
0 P(X=0) = P(HH) = .25
1 P(X=1) = P(HT or TH) = .5
2 P(X=0) = P(HH) = .25
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Draw a graph representing the distribution of X (# of tails in 2 flips)
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Some Terms to Know
Random Experiment
Random Variable
Discrete Random Variable Continuous Random Variable
Probability Distribution
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Terms
Random Experiment:
Examples:
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Terms Continued
Random Variable:
Examples
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Terms Continued
Discrete Random Variable
Example
Continuous Random Variable
Example
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Terms Continued
The Probability Distribution of a random variable, X,
Example:
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X counts the number of tails in two flips of a coin
x Probability of getting “x”
0 P(X=0) = .25
1 P(X=1) = .50
2 P(X=2) = .25
Specify the random experiment & the random variable for this probability distribution.
Is the RV discrete or continuous?
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Properties of Discrete Probability Distributions
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Mean of a Discrete RV
Mean value =
Example: X counts the number of tails showing in two flips of a fair coin Mean =
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Example: Your Turn
Example # 12, parental involvement
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Overview
Some Important Concepts/Definitions Associated with Probability Distributions
Discrete Distribution Example: Binomial Distribution More practice with counting and complex
probabilities Continuous Distribution Example:
Normal Distribution
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Binomial Distribution
If X counts the number of successes in a binomial experiment, then X is said to be a binomial RV. A binomial experiment is a random experiment that satisfies the following
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Binomial Example
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What is the Binomial Probability Distribution?
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Binomial Distribution
Let X count the number of successes in a binomial experiment which has n trials and the probability of success on any one trial is represented by p, then
Check for the last example: P(X = 2) = ____
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Mean of a Binomial RV
Example: Test guessing
In general: mean = Variance =
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Using the TI-84
To find P(X=a) for a binomial RV for an experiment with n trials and probability of success p
Binompdf(n, p, a) = P(X=a)
Binomcdf(n, p, a) = P(X <= a)
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Pascal’s Triangle & Binomial Coefficients
Handout
Pascal’s Triangle Applet http://www.mathforum.org/dr.cgi/pascal.cgi
?rows=10
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Using Tree Diagrams for finding Probabilities of Complex Events
For a one-clip paper airplane, which was flight-tested with the chance of throwing a dud (flies < 21 feet) being equal to 45%. What is the probability that exactly one of
the next two throws will be a dud and the other will be a success?
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Airplane Example
Source: NCTM Standards for Prob/Stat. D:\Standards\document\chapter6\data.htm
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Airplane Problem
A: Probability =
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Homework
Blood type problem Handout # 22, 26, 37
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Overview
Some Important Concepts/Definitions Associated with Probability Distributions
Discrete Distribution Example: Binomial Distribution More practice with counting and complex
probabilities Continuous Distribution Example:
Normal Distribution
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Continuous Distributions
Probability Density Function
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Example: Normal Distribution
Draw a picture Show Probabilities Show Empirical Rule
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What is Represented by a Normal Distribution?
Yes or No Birth weight of babies born at 36 weeks Time spent waiting in line for a roller
coaster on Sat afternoon? Length of phone calls for a give person IQ scores for 7th graders SAT scores of college freshman
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Penny Ages
Collect pennies with those at your table. Draw a histogram of the penny ages Describe the basic shape Do the data that you collected follow the
empirical rule?
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Penny Ages Continued
Based on your data, what is the probability that a randomly selected penny is is between 5 & 10 years old? Is at least 5 years old? Is at most 5 years old? Is exactly 5 years old? Find average penny age & standard
deviation of penny age
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Using your calculator
Normalcdf ( a, b, mean, st dev)
Use the calculator to solve problems on the previous page.
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Homework
Handout #’s 12, 14, 15, 16, 24