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# chapter4 - Continuous Probability Distributions Continuous...

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Continuous Probability Distributions

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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 σ σ σ μ μ μ μ μ μ μ μ μ μ μ μ μ μ σ μ 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

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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 (+) 0 830 1192 Capital Cost (-) -193K -180K -162K Annual Cost (-) -7000 -6600 -6000 Operation/Maint(-) -2192 -2049 -1742

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Example – Cost/Benefit Analysis of Sprewell- Bluff Project (III) (Flood Control, in \$100K) < - - = ) 0 . 15 , 5 . 8 ( elsewhere 0 0 . 15 0 . 12 0 . 3 / ) 0 . 15 ( 0 . 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 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 Sprewell-Bluff Project (IV) (Flood Control) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 ) ( 12 15 12 5 . 19 30 216 5385 . 75 . 9 2 12 ) 12 ( 15 2 15 5385 . 2 15 ) 75 . 9 / 1 ( 5385 . 75 . 9 ) 15 ( ) 12 ( ) ( 15 12 75 . 22 5 . 8 17 375 . 11 5 . 8 2 5 . 8 5 . 8 2 5 . 8 2 ) 375 . 11 / 1 ( 375 . 11 ) 50 . 8 ( ) ( 12 5 . 8 0 ) ( 5 . 8 elsewhere 0 0 . 15 0 . 12 75 . 9 / ) 0 . 15 ( 0 . 12 5 . 8 375 . 11 ) 5 . 8 ( ) ( 2 2 2 12 2 12 2 2 2 2 2 5 . 8 2 5 . 8 = - + - + = = - - - + = = - + = - + = + - = - - - = = - = - = = < - - = y F y y y y y y t t dt t F y F y y y y y t t dt t y F y y F y y y y y y f y y y y

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Example – Cost/Benefit Analysis of Sprewell-Bluff Project (V) (Flood Control) - + - + - < = - - = 15 1 15 12 051282 . 0 538462 . 1 538462 . 10 12 5 . 8 043956 . 0 747253 . 0 175824 . 3 5 . 8 0 ) ( elsewhere 0 0 . 15 0 . 12 75 . 9 / ) 0 . 15 ( 0 . 12 5 . 8 375 . 11 ) 5 .
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