chapter4 - Continuous Probability Distributions Continuous...

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Unformatted text preview: Continuous Probability Distributions Continuous Random Variables and Probability Distributions Random Variable: Y Cumulative Distribution Function (CDF): F ( y )=P( Y y ) Probability Density Function (pdf): f ( y )=d F ( y )/d y Rules governing continuous distributions: f ( y ) 0 2200 y P(a Y b) = F (b)- F (a) = P( Y =a) = 0 2200 a b a dy y f ) ( 1 ) ( = - dy y f Expected Values of Continuous RVs [ ] [ ] ( 29 [ ] [ ] [ ] [ ] ( 29 [ ] ( 29 a a Y V a dy y f y a dy y f a ay dy y f b a b ay b aY E b aY E b aY V b a b a dy y f b dy y yf a dy y f b ay b aY E Y E Y E dy y f dy y yf dy y f y dy y f y y dy y f y Y E Y E Y V dy y f y g Y g E dy y yf Y E b aY = = =- =- = = +- + = +- + = + + = + = = + = + = +- = +- = = +- = +- = =- =- = = = = = + - - - - - - - - - - - - - 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( 2 ) ( ) ( 2 ) ( ) ( 2 ) ( ) ( )) ( ( ) ( : Variance ) ( ) ( ) ( e) convergenc absolute (assuming ) ( ) ( : Value Expected Example Cost/Benefit Analysis of Sprewell-Bluff Project (I) Subjective Analysis of Annual Benefits/Costs of Project (U.S. Army Corps of Engineers assessments) Y = Actual Benefit is Random Variable taken from a triangular distribution with 3 parameters: A=Lower Bound (Pessimistic Outcome) B=Peak (Most Likely Outcome) C=Upper Bound (Optimistic Outcome) 6 Benefit Variables 3 Cost Variables Source: B.W. Taylor, R.M. North(1976). The Measurement of Uncertainty in Public Water Resource Development, American Journal of Agricultural Economics , Vol. 58, #4, Pt.1, pp.636-643 Example Cost/Benefit Analysis of Sprewell-Bluff Project (II) ($1000s, rounded) Benefit/Cost Pessimistic (A) Most Likely (B) Optimistic (C) Flood Control (+) 850 1200 1500 Hydroelec Pwr (+) 5000 6000 6000 Navigation (+) 25 28 30 Recreation (+) 4200 5400 7800 Fish/Wildlife (+) 57 127 173 Area Redvlp (+) 830 1192 Capital Cost (-)-193K-180K-162K Annual Cost (-)-7000-6600-6000 Operation/Maint(-)-2192-2049-1742 Example Cost/Benefit Analysis of Sprewell- Bluff Project (III) (Flood Control, in $100K) < - - = ) . 15 , 5 . 8 ( elsewhere . 15 . 12 . 3 / ) . 15 ( . 12 5 . 8 5 . 3 ) 5 . 8 ( ) ( y y y y k y y k y f Triangular Distribution with: lower bound=8.5 Peak=12.0 upper bound=15.0 Choose k 220d area under density curve is 1: Area below 12.0 is: 0.5((12.0-8.5) k ) = 1.75 k Area above 12.0 is 0.5((15.0-12.0) k ) = 1.50 k Total Area is 3.25 k k =1/3.25 Triangular Distribution (Not Scaled) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 16 Flood Control Benefits ($100K) Probability Density Example Cost/Benefit Analysis of...
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chapter4 - Continuous Probability Distributions Continuous...

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