Risk Assessment and Decision Analysis with Bayesian Networks

483

Appendix E: Statistical Distributions

E.1 Introduction

The continuous and discrete statistical distributions that are supported in AgenaRisk and approximated

using dynamic discretization are listed in Table E.1. Each distribution function is described more fully in

the following subsections. Note that:

1. AgenaRisk does not check for sensible bounds or parameters on any of these distributions. Be

careful to ensure you set the Bayesian network (BN) interval bounds to match those required

for the distribution. If you choose a value outside of the logical bounds for a parameter,

AgenaRisk will attempt to enforce the logical maximum or minimum value. For example,

the Chi-square distribution has a parameter called “degrees of freedom”—values of less than

one for this parameter are invalid, so if you choose a value less than one AgenaRisk will set

the value to the nearest valid value, which is one, automatically. However, despite this there

remains a danger that you may generate a probability density function that does not lie in the

state range dened for the node in question. When this happens an error message appears

warning you that zero cells have been generated and this will lead to an inconsistency upon

calculation of the risk model.

2. Similarly, when specifying a distribution that does not include zero, that is, X > 0, you should

take care to set a very small number close to zero as the lower bound. Be careful if you then

use a deterministic conditional function, such as 1/X, as a child node of this node since this

can lead to an overow or very long tail intervals on the resulting histogram of the marginal

distribution.

3. AgenaRisk allows continuous functions to be used on Integer Interval nodes and discrete func-

tions to be applied on Continuous Interval nodes. In the case of the Uniform function the formula-

tion used differs in each case.

4. The truncated Normal (TNormal) distribution will not necessarily have the mean and variance

that you specify. However, if the variance is small and the mean you specify is not relatively

close to zero for the range you specify, then the resulting distribution will have a mean close

to what you specied (this is because the resulting TNormal will be “almost” the same as a

Normal).

484 Appendix E: Statistical Distributions

E.2 Beta Distribution

Probability function:

VX r()

rm r

m

()

2

=

−

Domain: 0 ≤ X ≤ 1

Parameter domain(s): α > 0, β > 0

Mean: EX()

=

α

α+β

Variance: VX()

()

(1

)

2

=

αβ

α+βα+β+

Note: The domain of the Beta distribution can be extended to any nite range in the region L ≤ X ≤ U.

Example: Beta(3, 7, 0, 10)

E.3 Gamma Distribution

Probability function: PX x()

e

1

()

x

=

α−

β

Γα

α−β

Domain: X > 0

Parameter domain(s): α > 0, β > 0 where

1

β=

λ

and λ is a rate parameter.

Mean: E(X) = αβ

Variance: V(X) = αβ

2

Example: Gamma(3, 20)

Table E.1

Statistical Distributions

Student-t Chi-Square

Continuous Uniform Exponential

Beta Normal

Gamma Truncated Normal

Weibull Extreme Value

Logistic LogNormal

Triangular Binomial

Integer Uniform Hypergeometric

Negative Binomial Geometric

Poisson

0.014

0.012

0.01

0.0080

0.0060

0.0040

0.0020

Gamma(3, 20)

0.0

6.1608 26.16146.18166.16186.181106.16126.16146.16166.16

FIGURE E.1 Beta(3,7,0,10) distribution example.

485Appendix E: Statistical Distributions

E.4 Exponential Distribution

Probability function: P(X) = e

−λ x

Domain: X > 0

Parameter domain(s): λ > 0

Mean: E(X) = λ

Variance: V(X) = λ

2

Example: Exponential(2)

E.5 Normal Distribution

Probability function: =

σπ

−−

µσ

PX e()

x

1

2

()/(

2)

22

Domain: –∞ < X < ∞

Parameter domain(s): –∞ < μ < ∞, σ

2

> 0

Mean: E(X) = μ

Variance: V(X) = σ

2

Example: Normal(0, 100)

0.014

0.012

0.01

0.0080

0.0060

0.0040

0.0020

Gamma(3, 20)

0.0

6.1608 26.16146.18166.16186.181106.16126.16146.16166.16

FIGURE E.2 Gamma(3,20) distribution example.

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

Exponential (2)

0.0

0.0025789 0.40258 0.80258 1.2026 1.6026 2.0028 2.4026

FIGURE E.3 Exponential(2) distribution example.

486 Appendix E: Statistical Distributions

E.6 Truncated Normal Distribution

Probability function: PX e()

x

1

2

()/(

2)

22

=

σπ

−−

µσ

Domain: L ≤ X ≤ U

Parameter domain(s): –∞ < μ <∞, σ

2

> 0

Mean: E(X) ≈ μ

Variance: V (X) ≈ σ

2

Note: The domain of the (doubly) truncated Normal distribution is restricted to the region L ≤ X ≤ U

and under these circumstances the mean and variance of the truncated distribution is only approx-

imated by the mean and variance of the untruncated distribution. Depending on the truncation the

true mean and variance may differ signicantly from the supplied values.

Example: TNormal(0, 100, 0, 50)

E.7 LogNormal Distribution

Probability function: PX e()

x

x

1

2

(In)/(

2)

22

=

σπ

−−

µσ

Domain: X > 0

Parameter domain(s): –∞ < μ < ∞, σ

2

> 0

Mean: E(X) = e

(μ + (1/2σ

2

))

Variance: V(X) = e

2μ

e

σ

2

(

e

σ

2

– 1)

Example: LogNormal(1.5, 2)

Normal (0, 100)

0.04

0.036

0.032

0.028

0.024

0.02

0.016

0.012

0.0080

0.0040

0.0

–31.329–23.329 –15.329 –7.3287 0.67126 8.6713 16.67124.671

FIGURE E.4 Normal(0,100) distribution example.

TNormal (0, 100, 0, 50)

0.08

0.072

0.064

0.056

0.048

0.04

0.032

0.024

0.016

0.0080

0.0

0.062692 4.06278.062712.06316.06320.06324.063

FIGURE E.5 Truncated Normal (0,100,0,50) distribution example.