3_RandomVariable_20101018

3_RandomVariable_20101018 - i i i 2 Chapter 3 The Basic...

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i i i 2 Chapter 3 The Basic Concept of Univariate Variables 3 The Basic Concept of Univariate Variables Chapter Outline 3.1 Introduction 3.2 Random Variables (R.V.) 3.2.1 DeFnition of R.V. 3.2.2 Representing Events by R.V. 3.2.3 Types of R.V. 3.2.4 Revisiting R.V. 3.3 The Cumulative Distribution ±unction (cdf) 3.3.1 DeFnition and Properties of CD± 3.3.2 Determining P(Event) via CD±s 3.3.3 Graphing cdfs 3.4 The Distribution ±unction (df) 3.4.1 Discrete D± 3.4.2 The Continuous D± 3.4.3 Determining P(Event) via df 3.5 Characteristics of the D± 3.5.1 Location Parameters 3.5.2 Variability 3.5.3 The Probability of Tail Events 3.5.4 Quantile and Percentile 3.5.5 Moments 3.6 Revisiting Probability Models 3.6.1 Identical Random Variables vs. Identical Distributions 3.6.2 Distribution Shapes 3.6.3 Revisiting the Plot of cdf 3.6.4 ±rom (S, D ,P)to( R , B , F ) 3.6.5 The Metaphor of a Model House 3.6.6 Constructing a Probability Model 3.7 Summary Exercises
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i i i 3.1 Introduction 3 3.1 Introduction random variable (r.v.) discrete r.v. continuous r.v. cumulative distribution function (cdf) distribution function (df) probability mass function (pmf) cumulative distribution function (cdf) distribution function (df) probability density function (pdf) Random Variable Type Probability Function (cdf and df) mixed r.v. 3.1: Three types of random variable 3.2 Random Variables (R.V.) 3.2.1 De±nition of Random Variables Defnition 3.1 A random variable is a function that maps each sample point in the sample experiment to a real number. Points to keep in mind: 1. The random variable is a function. 2. The domain of an R.V. is a sample space denoted by S; the codomain is real numbers denoted by R . Random Variables are denoted by capital letters such as X and Y . The value of an R.V. is then denoted by the corresponding lower case letter, such as x and y . The relationship between X and x is giveninEquation(3.1): X(s) = x,s S ,x R (3.1) The relationship among X ,S , R , s ,and x is illusatrated in Figure 3.2.
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i i i 4 Chapter 3 The Basic Concept of Univariate Variables S s X x = X(s) R 3.2: The relationship among X ,S, R , s ,and x Example 3.1 Let X be the outcome of Fipping a fair die once. Explain the rela- tionship among X, s, and . In particular, explain whether or not the values of s and x are identical. Example 3.2 ±lip a fair coin once. Let X(H) = 1 ,X(T) = 0. Graph the relationship among X, s, and . Use the random variable X to represent the event of showing Heads and the event of showing Tails, respectively. S X( Head ) = 1 Tail ) = 0 R Head 3.3: ±lip a fair coin once
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i i i 3.2 Random Variables (R.V.) 5 Example 3.3 Flip a fair coin three times. We are interested in the number of events of showing Heads. Use an appropriate random variable to represent the event of showing two Heads? S E R X(( H, H, H )) = 3 3.4: Flip a coin three times
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i i i 6 Chapter 3 The Basic Concept of Univariate Variables Example 3.4 Consider a gambling game in which the host Fips a fair coin and the total money the player wins is related to when the ±rst Head appear. If Heads appears at the ±rst Fip, the player wins $1. If
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3_RandomVariable_20101018 - i i i 2 Chapter 3 The Basic...

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