note9_DataDistributionsIV_print(4).pdf

note9_DataDistributionsIV_print(4).pdf - Topic 9 Data...

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Topic 9. Data Distributions: Part IV Continuous Data Models Text Reference: Ravishanker’s Chapter 3 Reading Assignment: Ravishanker’s Sections 3.7-3.9 1/27
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Exponential Distribution Density Function A random variable Y is said to have an exponential distribution with parameter λ > 0 if it has density f ( y ) = ( λ e - λ y , if y > 0 , 0 , otherwise . Notation If Y has an exponential distribution with parameter λ , we write Y Exp( λ ) . Distribution Function If Y Exp( λ ), then the cdf is given by F ( y ) = ( 1 - e - λ y , if y > 0 , 0 , otherwise . 2/27
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Exponential Distribution Figure 9.1: The Exponential Distribution. 3/27
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Moment Generating Function If Y Exp( λ ), then the mgf is M Y ( t ) = E [ e tY ] = Z 0 e ty λ e - λ y dy = Z 0 λ e - ( λ - t ) y dy = λ λ - t , for t < λ . Moments Expanding in powers of t , we get M Y ( t ) = λ λ - t = 1 1 - t = X k =0 ( t ) k = X k =0 k ! λ k · t k k ! . Thus, E [ Y k ] = k ! λ k , for k = 0 , 1 , 2 , . . . . In particular, E [ Y ] = 1 λ , and Var( Y ) = E [ Y 2 ] - ( E [ Y ]) 2 = 2 λ 2 - 1 λ 2 = 1 λ 2 . 4/27
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Survival Function If Y Exp( λ ), then the survival function, or the tail-probability function, or the reliability function, is given by S ( y ) = 1 - F ( y ) = ( e - λ y , if y > 0 , 1 , otherwise . Hazard Function The hazard function of the exponential distribution is h ( y ) = f ( y ) S ( y ) = λ e - λ y e - λ y = λ. Thus, the exponential distribution has a constant hazard over time. That is, if Y Exp( λ ) is survival time (or time until “failure” or “death”), the survival function S ( y ) is the probability of survival beyond time y , and the hazard (instantaneous “failure” or “death” rate) is constant. A constant hazard rate means that the lifetimes are memoryless . 5/27
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Memoryless Property If Y Exp( λ ), then P ( Y > s + t | Y > t ) = P ( Y > s ) for all t > 0 and s > 0. Proof : If Y Exp( λ ), then for any t > 0 and s > 0, P ( Y > s + t | Y > t ) = P ( Y > s + t , Y > t ) P ( Y > t ) = P ( Y > s + t ) P ( Y > t ) = S ( s + t ) S ( t ) = e - λ ( s + t ) e - λ t = e - λ s = S ( s ) = P ( Y > s ) .
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