project storm fury review a stochastic variable has the following probability distribution: values...

Project Storm Fury

ReviewReview

A stochastic variable has the following probability distribution:A stochastic variable has the following probability distribution:

Values of XProbability distribution

of X

x P(X=x)

$1 P(X=1) = 1/3

$2 P(X=2) = 1/3

$3 P(X=3) = 1/3

ReviewReview

• What is X’s cumulative probability distribution?

• What is its expected value (X=?)

• What is the Variance of X?

• What is its standard deviation?

• What is X’s cumulative probability distribution?

• What is its expected value (X=?)

• What is the Variance of X?

• What is its standard deviation?

ReviewReview

What is the Variance of X?VarX = xi

2 P(X=xi) - 2

= [1(1/3) + 22(1/3) + 32(1/3)] - 22

= (1/3 + 4/3 + 3) - 4

= 2/3

What is its standard deviation ()?X = SqrRoot(VarX) = (2/3)1/2

= .8165

What is the Variance of X?VarX = xi

2 P(X=xi) - 2

= [1(1/3) + 22(1/3) + 32(1/3)] - 22

= (1/3 + 4/3 + 3) - 4

= 2/3

What is its standard deviation ()?X = SqrRoot(VarX) = (2/3)1/2

= .8165

Total Property Damage ($ of 1969)

Total Property Damage ($ of 1969)

Maximum sustained winds over timeMaximum sustained winds over time

Alternative HypothesesAlternative Hypotheses

• H1, the “beneficial” hypothesis. The average effect of seeding is to reduce maximum sustained wind speed.

• H2, the “null” hypothesis. Seeding has no effect on hurricanes. No change is induced in maximum sustained wind speed.

• H3, the “detrimental” hypothesis. The average effect of seeding is to increase the maximum sustained wind speed.

• H1, the “beneficial” hypothesis. The average effect of seeding is to reduce maximum sustained wind speed.

• H2, the “null” hypothesis. Seeding has no effect on hurricanes. No change is induced in maximum sustained wind speed.

• H3, the “detrimental” hypothesis. The average effect of seeding is to increase the maximum sustained wind speed.

Mathematical expressionsMathematical expressions

• P(w' | H2) = P(w) = fN(100, 15.6)

• P(w' | H1) = ƒN(85, 18.6)

• P(w' | H3 ) = ƒN(110, 18.6)

• P(w' | H2) = P(w) = fN(100, 15.6)

• P(w' | H1) = ƒN(85, 18.6)

• P(w' | H3 ) = ƒN(110, 18.6)

Probability density function for Debbie resultsProbability density function for Debbie results

• P(69, 85 | H1) = 1.50 x 2.14 =3.21

• P(69, 85 | H2) = 0.372 x 1.64 = 0.61

• P(69, 85 | H3) = 0.195 X 0.886 = 0.173

P(H1 | 69, 85) = (3.21 x 1/3)/(3.21 x 1/3 + 0.61 x 1/3 + 0.173 x 1/3)

= .81

P(H2 | 69, 85) = .15

P(H3 | 69, 85) = .04

• P(69, 85 | H1) = 1.50 x 2.14 =3.21

• P(69, 85 | H2) = 0.372 x 1.64 = 0.61

• P(69, 85 | H3) = 0.195 X 0.886 = 0.173

P(H1 | 69, 85) = (3.21 x 1/3)/(3.21 x 1/3 + 0.61 x 1/3 + 0.173 x 1/3)

= .81

P(H2 | 69, 85) = .15

P(H3 | 69, 85) = .04

Prior probabilities - pre and post DebbiePrior probabilities - pre and post Debbie

• P(H1) = .15

• P(H2) = .75

• P(H3) = .10

• P(H1) = .15

• P(H2) = .75

• P(H3) = .10

• P(H1) = .49

• P(H2) = .49

• P(H3) = .02

• P(H1) = .49

• P(H2) = .49

• P(H3) = .02

.81(.15)/ [.81(.15) +.15(.75) + .04(.1)] = .51 .15(.75)/ [.81(.15) +.15(.75) + .04(.1)] = .47.04(.1)/ [.81(.15) +.15(.75) + .04(.1)] = .02

The Seeding DecisionThe Seeding Decision

Probabilities assigned to wind changes occurring in the 12 hours before hurricane landfall

Cumulative probability functions

Probabilities assigned to wind changes occurring in the 12 hours before hurricane landfall

Cumulative probability functions

Probabilities assigned to wind changes occurring in the 12 hours before hurricane landfall.

Discrete approximation for five outcomes.

Probabilities assigned to wind changes occurring in the 12 hours before hurricane landfall.

Discrete approximation for five outcomes.

Interval of changes in maximum sustained wind

Representative value in discrete approximation

(%) If seeded If not seeded

Increase of 25% or more 32 0.038 0.054

Increase of 10 to 25% 16 0.143 0.206

Little change, +10 to -10% 0 0.392 0.480

Reduction of 10 to 25% -16 0.255 0.206

Reduction of 25% or more -34 0.172 0.054

Probability that windchange will bewithin interval

The seeding decision for the nominal hurricaneThe seeding decision for the nominal hurricane

$21.7M

The expected value of perfect informationThe expected value of perfect information

The value of further testsThe value of further tests

ReviewReview

1. Decide whose benefits and costs count, and how much. This is typically referred to as determining standing.

2. Select the portfolio of alternative initiatives. 3. Catalog potential consequences and select measurement

indicators. 4. Predict quantitative consequences over the life of the project

for those who have standing.5. Monetize (attach cash values to) all the predicted

consequences. 6. Discount for time to find present values.7. Sum up benefits and Costs for each initiative and Perform

sensitivity analysis underlying key assumptions

1. Decide whose benefits and costs count, and how much. This is typically referred to as determining standing.

2. Select the portfolio of alternative initiatives. 3. Catalog potential consequences and select measurement

indicators. 4. Predict quantitative consequences over the life of the project

for those who have standing.5. Monetize (attach cash values to) all the predicted

consequences. 6. Discount for time to find present values.7. Sum up benefits and Costs for each initiative and Perform

sensitivity analysis underlying key assumptions

Adapting to Climate Change 1Adapting to Climate Change 1

Adapting to Climate Change 2Adapting to Climate Change 2

Adapting to Climate Change 3Adapting to Climate Change 3

Adapting to Climate Change 4Adapting to Climate Change 4

Adapting to Climate Change 5Adapting to Climate Change 5

Source: Oregon Environmental Quality Commission, Oregon Climate Change Adaptation Framework. December 10, 2010, Salem OR