Probability Statistics Continuous Random Variables Probability Density Function

Probability statistics continuous random variables

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Probability & Statistics Continuous Random Variables Probability Density Function The Mean Expected Value Rule The Variance Cumulative Distribution Functions Common Random Variables 4/24 1 Continuous Random Variables Probability Density Function The Mean Expected Value Rule The Variance 2 Cumulative Distribution Functions 3 Common Random Variables
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Probability & Statistics Continuous Random Variables Probability Density Function The Mean Expected Value Rule The Variance Cumulative Distribution Functions Common Random Variables 5/24 Continuous Random Variables Definition X : sample space Ω R outcome ω 7→ x
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Probability & Statistics Continuous Random Variables Probability Density Function The Mean Expected Value Rule The Variance Cumulative Distribution Functions Common Random Variables 5/24 Continuous Random Variables Definition X : sample space Ω R outcome ω 7→ x Notation : random variable X , numerical value x
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Probability & Statistics Continuous Random Variables Probability Density Function The Mean Expected Value Rule The Variance Cumulative Distribution Functions Common Random Variables 6/24 Probability Density Function
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Probability & Statistics Continuous Random Variables Probability Density Function The Mean Expected Value Rule The Variance Cumulative Distribution Functions Common Random Variables 6/24 Probability Density Function Definition A random variable X is called continuous if there is a nonnegative function f X , called the probability density function of X , such that P ( a X b ) , Z b a f X ( x ) dx
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Probability & Statistics Continuous Random Variables Probability Density Function The Mean Expected Value Rule The Variance Cumulative Distribution Functions Common Random Variables 7/24 Probability Density Function Properties 1 f X ( x ) 0 for all x
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Probability & Statistics Continuous Random Variables Probability Density Function The Mean Expected Value Rule The Variance Cumulative Distribution Functions Common Random Variables 7/24 Probability Density Function Properties 1 f X ( x ) 0 for all x 2 Z -∞ f X ( x ) dx = 1
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Probability & Statistics Continuous Random Variables Probability Density Function The Mean Expected Value Rule The Variance Cumulative Distribution Functions Common Random Variables 7/24 Probability Density Function Properties 1 f X ( x ) 0 for all x 2 Z -∞ f X ( x ) dx = 1 3 For any subset B of the real line P ( X B ) = Z B f X ( x ) dx
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Probability & Statistics Continuous Random Variables Probability Density Function The Mean Expected Value Rule The Variance Cumulative Distribution Functions Common Random Variables 8/24 The Mean Mean The expected value ( also called the expectation or the mean) of a random variable X , with PDF f X , is defined by E [ X ] , Z -∞ xf X ( x ) dx
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Probability & Statistics Continuous Random Variables Probability Density Function The Mean Expected Value Rule The Variance Cumulative Distribution Functions Common Random Variables 9/24 Linearity of Expectations Linearity of Expectations Given a random variable X : E [ aX + b ] = a E [ X ] + b
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Probability & Statistics Continuous Random Variables Probability Density Function The Mean Expected Value Rule The Variance Cumulative Distribution Functions Common Random Variables 10/24 Expected Value Rule Let X be a random variable, and Y = g ( X ) be a function of X . By definition: E [ g ( X )] = E [ Y ] = Z -∞ yf Y ( y ) dy
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Probability & Statistics Continuous Random Variables Probability Density Function The Mean Expected Value Rule
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  • Normal Distribution, Probability theory, probability density function, CDF Properties

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