Lect9_2700_s09 - ENGRD 2700 Basic Engineering Probability...

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Expectation (cont) Densities to memorize . . . Normal Gamma Density Title Page JJ II J I Page 1 of 24 Go Back Full Screen Close Quit ENGRD 2700 Basic Engineering Probability and Statistics Lecture 9: Densities David S. Matteson School of Operations Research and Information Engineering Rhodes Hall, Cornell University Ithaca NY 14853 USA [email protected] February 16, 2009
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Expectation (cont) Densities to memorize . . . Normal Gamma Density Title Page JJ II J I Page 2 of 24 Go Back Full Screen Close Quit 1. Expectation & Variance (continued) Definitions are the same as for the discrete case, except now we inte- grate instead of sum. Suppose X has pdf f ( x ) E ( X ) = Z -∞ xf ( x ) dx = μ E ( h ( X )) = Z -∞ h ( x ) f ( x ) dx E ( X 2 ) = Z -∞ x 2 f ( x ) dx Var( X ) = Z -∞ ( x - μ ) 2 dx = E ( X - μ ) 2 = E ( X 2 ) - ( E ( X )) 2 . Hints for calculating: 1. If P [ X 0] = 1, then E ( X ) = Z 0 (1 - F ( x )) dx.
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Expectation (cont) Densities to memorize . . . Normal Gamma Density Title Page JJ II J I Page 3 of 24 Go Back Full Screen Close Quit Example: Exponential density. Here F ( x ) = 1 - e - λx , x > 0 . Therefore E ( X ) = Z 0 (1 - F ( x )) dx = Z 0 e - λx dx = 1 λ . What is the variance? Write E ( X 2 ) = Z 0 P [ X 2 > x ] dx = Z 0 P [ X > x ] dx = 2 λ 2 Therefore Var( X ) = Conclude for an exponential density with parameter λ > 0: E ( X ) = 1 λ , Var( X ) = 1 λ 2
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Expectation (cont) Densities to memorize . . . Normal Gamma Density Title Page JJ II J I Page 4 of 24 Go Back Full Screen Close Quit 2. Densities to memorize (hint hint) 1. Exponential density with parameter λ > 0 . Facts: Density: f ( x ) = ( λe - λx , if x > 0 , 0 , if x < 0 . Distribution function: F ( x ) = ( 1 - e - λx , if x > 0 , 0 , if x < 0 . Distribution tail: 1 - F ( x ) = P [ X > x ] = e - λx , x > 0 , where X has the exponential density. Mean and variance: E ( X ) = 1 λ , Var( X ) = 1 λ 2 .
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Expectation (cont) Densities to memorize . . . Normal Gamma Density Title Page JJ II J I Page 5 of 24 Go Back Full Screen Close Quit Memory-less property: for x > 0 , t > 0 P [ X > t + x | X > t ] = P [ X > x ] . Reason: see blackboard. Interpretation: If X is the failure time of a component, given that the component survives at least t units, the probability that it survives another x units is that same as if the component were fresh and had no history. Constant hazard function. The hazard function associated with density f and distribution function F is h ( x ) = f ( x ) 1 - F ( x ) = density( x ) tail( x ) . For the exponential density h ( x ) = λ, x > 0 . Interpretation: Compute for x > 0, u > 0, u small,: P [ X ( x, x + u ] | X > x ] = R x + u x f ( s ) ds 1 - F ( x ) = hf ( ζ ) 1 - F ( x ) hf ( x ) 1 - F ( x )
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Expectation (cont) Densities to memorize . . . Normal Gamma Density Title Page JJ II J I Page 6 of 24 Go Back Full Screen Close Quit by the mean value theorem and assumed continuity of f . Recall x < ζ < x + u . This expression equals = uh ( x ) .
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