Revised Lecture 4 ECN 221 Spring 2008

# Revised Lecture 4 ECN 221 Spring 2008 - Random Variables...

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Random Variables and Their Probability Distributions: Learning Objectives 1. 1. Describe Describe Discrete Discrete Random Variables Random Variables 2. 2. Understand the relationship between a Understand the relationship between a random variable and its distribution random variable and its distribution 3. 3. Describe Describe Continuous Random Variables Continuous Random Variables 4. 4. Describe the Normal Distribution Describe the Normal Distribution 5. 5. Understand Understand Sampling Distributions Sampling Distributions 6. 6. Understand the importance of the central Understand the importance of the central limit theorem limit theorem

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Discrete Random Variables 1. 1. A discrete random variable is a A discrete random variable is a numerical outcome of an experiment numerical outcome of an experiment Example: Number of tails in 2 coin toss Example: Number of tails in 2 coin toss Discrete Discrete numbers (0, 1, 2, 3, etc.) numbers (0, 1, 2, 3, etc.) Often Often obtained by counting obtained by counting
Discrete Random Variable Examples Experiment Random Variable Possible Values Count Cars at Toll Between 11:00 & 1:00 # Cars Arriving 0, 1, 2, ..., Make 100 Sales Calls # Sales 0, 1, 2, ..., 100 Inspect 70 Radios # Defective 0, 1, 2, ..., 70 Answer 33 Questions # Correct 0, 1, 2, ..., 33

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Continuous Random Variable A continuous random variable is usually a numerical outcome of an experiment Weight of a student (e.g., 115, 156, etc.) Whole or fractional number Obtained by measuring Infinite number of values in interval Too many possibilities to list
Continuous Random Variable Examples Measure Time Between Arrivals Inter-Arrival Time 0, 1.3, 2.78, ... Experiment Random Variable Possible Values Weigh 100 People Weight 45.1, 78, ... Measure Part Life Hours 900, 875.9, ... Amount spent on food \$ amount 54.12, 42, ...

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The Probability Distribution of a discrete random variable 1. It is a list of all possible [ x , p ( x )] pairs x = value of random variable (outcome) p ( x ) = probability associated with value 2. Mutually exclusive (no overlap) 3. Collectively exhaustive (nothing left out) 4. 0 p ( x ) 1 for all x 1. Σ p ( x ) = 1
Discrete Probability Distribution Probability Distribution Values, x Probabilities, p ( x ) 0 1/4 = .25 1 2/4 = .50 2 1/4 = .25 Experiment: Toss 2 coins. Count number of tails. © 1984-1994 T/Maker Co.

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Visualizing Discrete Probability Distributions Listing Table Formula # Tails f( x ) Count p( x ) 0 1 .25 1 2 .50 2 1 .25 p x n x!(n – x)! ( ) ! = p x (1 – p) n - x Graph .00 .25 .50 0 1 2 x p( x ) { (0, .25), (1, .50), (2, .25) }
Summary Measures of a Discrete Probability Distribution Expected Value (Mean of probability distribution) Weighted average of all possible values μ = E ( x ) = Σ x p ( x ) Variance Weighted average of squared deviation about mean σ 2 = E [( x - μ 29 2 ] = Σ ( x - μ 29 2 p ( x ) Standard Deviation 2 σ σ =

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Summary Measures Calculation Table x p(x) x p(x) x – μ Total Σ ( x - μ 29 2 p ( x ) (x – μ29 2 (x – μ29 2 p(x) Σ x p ( x )
Thinking Challenge You toss 2 coins. You’re interested in the number of tails. What are the expected value , variance , and standard deviation of this random variable, number of tails?

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