Appendix - Distributions

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DISTRIBUTIONS

512

18.2

Continuous univariate distributions

Table 18.1 Beta density: Beta(, ) Model p() =1 B(,)

Examples 1 (1 )1()() (+)Density

3.0

= 4, = 1 = 0.2, = 6

2.0

with B(, ) =

2.5

= 3, = 3 = 0.7, = 0.7

1.5

Condition: > 0, > 0 Range: [0,1] Parameters: , : shape Moments mean: mode: variance: (+) 1 (+2) (+)2 (++1)

0.0

0.5

1.0

= 1, = 1

0.0

0.2

0.4

0.6

0.8

1.0

Program commands R: dbeta(theta,alpha,beta)

WB/JAGS: theta ~ dbeta(alpha,beta) SAS: theta ~ beta(alpha,beta)

DISTRIBUTIONS Table 18.2 Cauchy distribution: Cauchy(, ) Model p() = (1 2 +()2

513

Examples )0.5 0.4 N( = 5, = 1)

Condition: > 0 Range: (, ) Parameters: : location, : scale

Density

0.2

0.3

= 5, = 1

= 7, = 1

= 4, = 2 0.1 0.0 0

2

4

6

8

10

Moments mean: mode: variance: -

Program commands R: WB/JAGS: SAS: dcauchy(theta,mu,sigma) theta ~ cauchy(mu,sigma)

Note: Cauchy distribution is a special case of location-scale t-distribution: Cauchy(, ) = t(1, , ).

DISTRIBUTIONS Table 18.3 Chi-squared density: 2 () Model p() =1 (/2)1 e/2 (/2)2/21.0 =1 0.8

514

Examples

Condition: > 0 Range: = 2 : [0, ) otherwise : (0, ) Parameters: : degrees of freedom

Density

0.2

0.4

0.6

= 0.01

=5 = 10

0.0 0

2

4

6

8

Moments mean: mode: nu 2 ( 2), otherwise

Program commands R: WB/JAGS: SAS: dchisq(theta,nu) theta ~ dchisqr(nu) theta ~ chisq(nu)

variance: 2

Note: Chi-squared is a special case of a gamma distribution: 2 () = Gamma( = /2, = 1/2) (rate). JAGS oers a non-central 2 -distribution: theta dnchisqr(nu,delta), > 0 non-centrality parameter. JAGS oers an F-distribution (ratio of 2 independent 2 s): theta df(nu1, nu2), with nu1, nu2 = dfs of numerator and denominator, resp.

DISTRIBUTIONS Table 18.4 Exponential density: Exp() Model rate: p() = eDensity

515

Examples3.0 =4 rate =

Condition: > 0 Range: [0, ) Parameters: : rate

1.0

1.5

2.0

2.5

=2 0.5

0.0

= 0.1 0

=1

2

4

6

8

Moments rate: mean: mode: variance: 1

Program commands R: dexp(theta,lambda)

01 2

WB/JAGS: theta ~ dexp(lambda) SAS: (scale) theta ~ expon(iscale=lambda) theta ~ expon(scale=ilambda)

Note: Exponential is special case of gamma distribution: Exp()= Gamma( = 1, ).

DISTRIBUTIONS Table 18.5 Gamma density: Gamma(, ) Model rate: p() = ()

516

Examples (1) e

3.0

= 0.1, = 0.1

1/scale =

Condition: > 0, > 0 Range: = 1 : (0, ) otherwise : [0, ) Parameters: : shape, : rate

2.0

2.5

= 20, = 20

Density

1.0

1.5

0.5

= 1, = 1 = 4, = 1

0.0 0

2

4

6

8

Moments rate: mean: mode: variance: 1 2

Program commands R: (scale) ( 1) WB/JAGS: SAS: (scale) dgamma(theta,alpha,rate=beta) dgamma(theta,alpha,scale=ibeta) theta ~ dgamma(alpha,beta) theta ~ gamma(alpha,iscale=beta) theta ~ gamma(alpha,scale=ibeta)

Note: WB and JAGS oer a generalized gamma distribution GenGamma: GenGamma(, , ) 1/ Gamma(, ), with = 1/ . WB/JAGS command: theta dgen.gamma(alpha,beta,lambda).

DISTRIBUTIONS Table 18.6 Inverse chi-squared density: Inv 2 () Model p() =1 (/2+1) e1/(2) (/2)2/24 =5

517

Examples

Condition: > 0 Range: (0, ) Parameters: : degrees of freedom

Density

1

2

3

=3 =1 0 0 5 10 15

Moments mean: mode: variance:1 2 1 +2

Program commands ( > 2) R: WB/JAGS: ( > 4) SAS: dchisq(1/theta,nu)/theta^2 theta 0, > 0 Range: (0, ) Parameters: : shape, : rate

Density

2

3

= 20, = 20

1

Gamma( = 4, = 1) 0 = 1, = 1 0 2 4 6 8

Moments rate: mean: mode: (1)

Program commands

R: (scale)

dgamma(1/theta,alpha,rate=beta)/theta^2 dgamma(1/theta,alpha,scale=beta)/theta^2

(+1)

WB/JAGS: theta 0 Range: (, ) Parameters: : location, : scale Examples

519

0.6

0.5

= 5, = 1

= 7, = 1

Density

0.3

0.4

= 4, = 2 0.2 0.0 0 0.1

2

4

6

8

10

Moments scale: mean: mode:

Program commands R: dlaplace(theta,mu,sigma)

WB/JAGS: (rate) theta ~ ddexp(isigma) SAS: (rate) theta ~ laplace(mu,scale=sigma) theta ~ laplace(mu,iscale=isigma)

variance: 2 2

Note: Laplace distribution is also called double exponential distribution. R function dlaplace is available from R package VGAM.

DISTRIBUTIONS Table 18.9 Logistic distribution: Logistic(, ) Model p() = ( )[ ( )]2 exp exp Examples

520

0.5

Condition: > 0 Range: (, ) Parameters: : location, : scale

Density

0.3

0.4

N( = 5, = 1)

= 5, = 1 0.2 = 7, = 1

0.0

0.1

= 4, = 2

0

2

4

6

8

10

Moments mean: mode: variance: 32 2

Program commands R: dlogis(theta,mu,sigma)

WB/JAGS: theta ~ dlogis(mu,isigma) (rate) SAS: theta ~ logistic(mu,sigma)

DISTRIBUTIONS Table 18.10 Lognormal distribution: LN(, 2 ) Model p() =1 2

521

Examples ( ) 2 exp (log()) 2 2

0.7

= 0, = 1

0.1

Condition: > 0 Range: (0, ) Parameters: : location, : scale

0.4

0.5

0.6

Density

= 2, = 1

0.3

0.2

= 4, = 2

0.0

= 0, = 2 0 2 4 6 8 10

Moments mean: mode: exp( + 2 ) exp( )2

Program commands R: dlnorm(theta,mu,sigma)

WB/JAGS: theta ~ dlnorm(mu,isigma2) SAS: theta ~ lognormal(mu,sd=sigma) theta ~ lognormal(mu,var=sigma2) theta ~ lognormal(mu,prec=isigma2)

variance: exp(2( + 2 )) exp(2 + 2 )

DISTRIBUTIONS Table 18.11 Normal distribution: N(, 2 ) Model p() =1 2

522

Examples ( ) 2 exp () 2 2

0.5

0.4

= 5, = 1 = 7, = 1

Condition: > 0 Range: (, ) Parameters: : location, : scale

Density

0.2

0.3

= 4, = 2

0.0 0

0.1

2

4

6

8

10

Moments mean: mode: 2

Program commands R: dnorm(theta,mu,sigma)

WB/JAGS: theta ~ dnorm(mu,isigma2) SAS: theta ~ normal(mu,sd=sigma) theta ~ normal(mu,var=sigma2) theta ~ normal(mu,prec=isigma2)

variance:

DISTRIBUTIONS Table 18.12 Location-scale Students t-distribution: t(, , ) Model p() =( +1 ) 2 ( ) 2

523

Examples ( 1+ ) +1 20.5 0.4 N( = 5, = 1)

()2 2

0.0

Condition: > 0, > 0 Range: (, ) Parameters: : location, : scale : degrees of freedom

Density

0.3

0.1

0.2

= 10, = 4, = 2

= 20, = 5, = 1

= 2, = 5, = 1

0

2

4

6

8

10

Moments mean: mode: variance: (if > 1) 2 2

Program commands R: dt(nu,(theta-mu)/sigma)/sigma

WB/JAGS: theta ~ dt(mu,isigma2,nu) (if > 2) SAS: theta ~ t(mu,sd=sigma,nu) theta ~ t(mu,var=sigma2,nu) theta ~ t(mu,prec=isigma2,nu)

DISTRIBUTIONS Table 18.13 Pareto distribution: Pareto(, ) Model p() =

524

Examples ( )+1 4 = 4, = 1

Condition: > 0, > 0 Range: (, ) Parameters: : shape, : location

Density

2

3

1

= 4, = 4

= 1, = 1 0 1

2

3

4

5

6

7

8

Moments mean: mode: variance: 1

Program commands (if > 1) R: dpareto(theta,beta,alpha)

2 (1)2 (2)

WB/JAGS: theta ~ dpareto(alpha,beta) (if > 2) SAS: theta ~ pareto(alpha,beta)

Note: R function dpareto is available from R package VGAM.

DISTRIBUTIONS Table 18.14 Scaled inverse chi-squared density: Inv 2 (, s2 ) Model p() =(/2)/2 (/2+1) s2 /(2) e (/2) s 0.8 = 5, s2 = 1 0.6

525

Examples

Condition: > 0, s > 0 Range: (0, ) Parameters: : degrees of freedom, s2 : scale

Density

0.4

= 3, s2 = 1 0.2

= 3, s2 = 5 0.0 0

5

10

15

Moments mean: mode: variance: 2 2 s 2 +2 s 2 2 4 (2)2 (4) s

Program commands ( > 2) R: WB/JAGS: ( > 4) SAS: dchisq(nu*s^2/theta,nu)nu* s^2/theta^2 theta 0, > 0 Range: = 1 : [0, ) otherwise : (0, ) Parameters: : shape, : scale

Density

0.6

0.8

0.2

0.4

= 2, = 2

= 2, = 4

0

2

4

6

8

10

Moments mean: mode: (1 + 1/) (1 1/)1/ (if > 1)

Program commands R: dweibull(theta,alpha,beta)

WB/JAGS: theta ~ dweib(alpha,ibeta) SAS: theta ~ weibull(0,alpha,beta)

variance: [ ] 2 (1 + 2/) 2 (1 + 2/)

Note: SAS: more general Weibull distribution with additional > 0 = lower limit of range: weibull(mu,alpha,beta), with / in Weibull distribution replaced by ( )/.

DISTRIBUTIONS Table 18.16 Uniform distribution: U(, ) Model p() =1

527

Examples1.2

0.0

0.2

Condition: > Range: [, ] Parameters: : lower limit, : upper limit

Density

0.4

0.6

0.8

1.0

= 1, = 2

=