## Stata 15 help for probfun

```
[FN] Statistical functions

Functions

The probability distribution and density functions are organized under

Beta and noncentral beta distributions
Binomial distribution
Cauchy distribution
Chi-squared and noncentral chi-squared distributions
Dunnett's multiple range distribution
Exponential distribution
F and noncentral F distributions
Gamma distribution
Hypergeometric distribution
Inverse Gaussian distribution
Laplace distribution
Logistic distribution
Negative binomial distribution
Normal (Gaussian), binormal, and multivariate normal distributions
Poisson distribution
Student's t and noncentral Student's t distributions
Tukey's Studentized range distribution
Weibull distribution
Weibull (proportional hazards) distribution
Wishart distribution

Beta and noncentral beta distributions

Description:  the probability density of the beta distribution, where
a and b are shape parameters; 0 if x < 0 or x > 1
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain x:     -8e+307 to 8e+307; interesting domain is 0 < x < 1
Range:        0 to 8e+307

ibeta(a,b,x)
Description:  the cumulative beta distribution with shape parameters a
and b; 0 if x < 0; 1 if x > 1

ibeta() returns the regularized incomplete beta
function, also known as the incomplete beta function
ratio.  The incomplete beta function without
regularization is given by
(gamma(a)*gamma(b)/gamma(a+b))*ibeta(a,b,x)
or, better when a or b might be large,
exp(lngamma(a)+lngamma(b)-lngamma(a+b))*ibeta(a,b,x).

Here is an example of the use of the regularized
incomplete beta function.  Although Stata has a
cumulative binomial function (see binomial()), the
probability that an event occurs k or fewer times in n
trials, when the probability of one event is p, can be
evaluated as cond(k==n,1,1-ibeta(k+1,n-k,p)).  The
reverse cumulative binomial (the probability that an
event occurs k or more times) can be evaluated as
cond(k==0,1,ibeta(k,n-k+1,p)).
Domain a:     1e-10 to 1e+17
Domain b:     1e-10 to 1e+17
Domain x:     -8e+307 to 8e+307; interesting domain is 0 < x < 1
Range:        0 to 1

ibetatail(a,b,x)
Description:  the reverse cumulative (upper tail or survivor) beta
distribution with shape parameters a and b; 1 if x < 0;
0 if x > 1

ibetatail() is also known as the complement to the
incomplete beta function (ratio).
Domain a:     1e-10 to 1e+17
Domain b:     1e-10 to 1e+17
Domain x:     -8e+307 to 8e+307; interesting domain is 0 < x < 1
Range:        0 to 1

invibeta(a,b,p)
Description:  the inverse cumulative beta distribution: if
ibeta(a,b,x) = p, then invibeta(a,b,p) = x
Domain a:     1e-10 to 1e+17
Domain b:     1e-10 to 1e+17
Domain p:     0 to 1
Range:        0 to 1

invibetatail(a,b,p)
Description:  the inverse reverse cumulative (upper tail or survivor)
beta distribution: if ibetatail(a,b,x) = p, then
invibetatail(a,b,p) = x
Domain a:     1e-10 to 1e+17
Domain b:     1e-10 to 1e+17
Domain p:     0 to 1
Range:        0 to 1

Description:  the probability density function of the noncentral beta
distribution; 0 if x < 0 or x > 1

a and b are shape parameters, np is the noncentrality
parameter, and x is the value of a beta random variable.

preferred function to use for the central beta
distribution. nbetaden() is computed using an algorithm
described in Johnson, Kotz, and Balakrishnan (1995).
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain np:    0 to 1,000
Domain x:     -8e+307 to 8e+307; interesting domain is 0 < x < 1
Range:        0 to 8e+307

nibeta(a,b,np,x)
Description:  the cumulative noncentral beta distribution; 0 if x < 0;
1 if x > 1

a and b are shape parameters, np is the noncentrality
parameter, and x is the value of a beta random variable.

nibeta(a,b,0,x) = ibeta(a,b,x), but ibeta() is the
preferred function to use for the central beta
distribution. nibeta() is computed using an algorithm
described in Johnson, Kotz, and Balakrishnan (1995).
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain np:    0 to 10,000
Domain x:     -8e+307 to 8e+307; interesting domain is 0 < x < 1
Range:        0 to 1

invnibeta(a,b,np,p)
Description:  the inverse cumulative noncentral beta distribution: if
nibeta(a,b,np,x) = p, then invnibeta(a,b,np,p) = x
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain np:    0 to 1,000
Domain p:     0 to 1
Range:        0 to 1

Binomial distribution

binomialp(n,k,p)
Description:  the probability of observing floor(k) successes in
floor(n) trials when the probability of a success on one
trial is p
Domain n:     1 to 1e+6
Domain k:     0 to n
Domain p:     0 to 1
Range:        0 to 1

binomial(n,k,p)
Description:  the probability of observing floor(k) or fewer successes
in floor(n) trials when the probability of a success on
one trial is p; 0 if k < 0; or 1 if k > n
Domain n:     0 to 1e+17
Domain k:     -8e+307 to 8e+307; interesting domain is 0 < k < n
Domain p:     0 to 1
Range:        0 to 1

binomialtail(n,k,p)
Description:  the probability of observing floor(k) or more successes
in floor(n) trials when the probability of a success on
one trial is p; 1 if k < 0; or 0 if k > n
Domain n:     0 to 1e+17
Domain k:     -8e+307 to 8e+307; interesting domain is 0 < k < n
Domain p:     0 to 1
Range:        0 to 1

invbinomial(n,k,p)
Description:  the inverse of the cumulative binomial; that is, the
probability of success on one trial such that the
probability of observing floor(k) or fewer successes in
floor(n) trials is p
Domain n:     1 to 1e+17
Domain k:     0 to n - 1
Domain p:     0 to 1 (exclusive)
Range:        0 to 1

invbinomialtail(n,k,p)
Description:  the inverse of the right cumulative binomial; that is,
the probability of success on one trial such that the
probability of observing floor(k) or more successes in
floor(n) trials is p
Domain n:     1 to 1e+17
Domain k:     1 to n
Domain p:     0 to 1 (exclusive)
Range:        0 to 1

Cauchy distribution

cauchyden(a,b,x)
Description:  the probability density of the Cauchy distribution with
location parameter a and scale parameter b
Domain a:     -1e+300 to 1e+300
Domain b:     1e-100 to 1e+300
Domain x:     -8e+307 to 8e+307
Range:        0 to 8e+307

cauchy(a,b,x)
Description:  the cumulative Cauchy distribution with location
parameter a and scale parameter b
Domain a:     -1e+300 to 1e+300
Domain b:     1e-100 to 1e+300
Domain x:     -8e+307 to 8e+307
Range:        0 to 1

cauchytail(a,b,x)
Description:  the reverse cumulative (upper tail or survivor) Cauchy
distribution with location parameter a and scale
parameter b

cauchytail(a,b,x) = 1 - cauchy(a,b,x)
Domain a:     -1e+300 to 1e+300
Domain b:     1e-100 to 1e+300
Domain x:     -8e+307 to 8e+307
Range:        0 to 1

invcauchy(a,b,p)
Description:  the inverse of cauchy(): if cauchy(a,b,x) = p, then
invcauchy(a,b,p) = x
Domain a:     -1e+300 to 1e+300
Domain b:     1e-100 to 1e+300
Domain p:     0 to 1 (exclusive)
Range:        -8e+307 to 8e+307

invcauchytail(a,b,p)
Description   the inverse of cauchytail(): if cauchytail(a,b,x) = p,
then invcauchytail(a,b,p) = x
Domain a:     -1e+300 to 1e+300
Domain b:     1e-100 to 1e+300
Domain p:     0 to 1 (exclusive)
Range:        -8e+307 to 8e+307

lncauchyden(a,b,x)
Description:  the natural logarithm of the density of the Cauchy
distribution with location parameter a and scale
parameter b
Domain a:     -1e+300 to 1e+300
Domain b:     1e-100 to 1e+300
Domain x:     -8e+307 to 8e+307
Range:        -1650 to 230

Chi-squared and noncentral chi-squared distributions

chi2den(df,x)
Description:  the probability density of the chi-squared distribution
with df degrees of freedom; 0 if x < 0
Domain df:    2e-10 to 2e+17 (may be nonintegral)
Domain x:     -8e+307 to 8e+307
Range:        0 to 8e+307

chi2(df,x)
Description:  the cumulative chi-squared distribution with df degrees
of freedom; 0 if x < 0

chi2(df,x) = gammap(df/2,x/2)
Domain df:    2e-10 to 2e+17 (may be nonintegral)
Domain x:     -8e+307 to 8e+307; interesting domain is x > 0
Range:        0 to 1

chi2tail(df,x)
Description:  the reverse cumulative (upper tail or survivor)
chi-squared distribution with df degrees of freedom; 1
if x < 0

chi2tail(df,x) = 1 - chi2(df,x)
Domain df:    2e-10 to 2e+17 (may be nonintegral)
Domain x:     -8e+307 to 8e+307; interesting domain is x > 0
Range:        0 to 1

invchi2(df,p)
Description:  the inverse of chi2(): if chi2(df,x) = p, then
invchi2(df,p) = x
Domain df:    2e-10 to 2e+17 (may be nonintegral)
Domain p:     0 to 1
Range:        0 to 8e+307

invchi2tail(df,p)
Description:  the inverse of chi2tail(): if chi2tail(df,x) = p, then
invchi2tail(df,p) = x
Domain df:    2e-10 to 2e+17 (may be nonintegral)
Domain p:     0 to 1
Range:        0 to 8e+307

nchi2den(df,np,x)
Description:  the probability density of the noncentral chi-squared
distribution; 0 if x <= 0

df denotes the degrees of freedom, np is the
noncentrality parameter, and x is the value of
chi-squared.

nchi2den(df,0,x) = chi2den(df,x), but chi2den() is the
preferred function to use for the central chi-squared
distribution.
Domain df:    2e-10 to 1e+6 (may be nonintegral)
Domain np:    0 to 10,000
Domain x:     -8e+307 to 8e+307
Range:        0 to 8e+307

nchi2(df,np,x)
Description:  the cumulative noncentral chi-squared distribution; 0 if
x < 0

df denotes the degrees of freedom, np is the
noncentrality parameter, and x is the value of
chi-squared.

nchi2(df,0,x) = chi2(df,x), but chi2() is the preferred
function to use for the central chi-squared
distribution.
Domain df:    2e-10 to 1e+6 (may be nonintegral)
Domain np:    0 to 10,000
Domain x:     -8e+307 to 8e+307; interesting domain is x > 0
Range:        0 to 1

nchi2tail(df,np,x)
Description:  the reverse cumulative (upper tail or survivor)
noncentral chi-squared distribution; 1 if x < 0

df denotes the degrees of freedom, np is the
noncentrality parameter, and x is the value of
chi-squared.
Domain df:    2e-10 to 1e+6 (may be nonintegral)
Domain np:    0 to 10,000
Domain x:     -8e+307 to 8e+307
Range:        0 to 1

invnchi2(df,np,p)
Description:  the inverse cumulative noncentral chi-squared
distribution: if nchi2(df,np,x) = p, then
invnchi2(df,np,p) = x
Domain df:    2e-10 to 1e+6 (may be nonintegral)
Domain np:    0 to 10,000
Domain p:     0 to 1
Range:        0 to 8e+307

invnchi2tail(df,np,p)
Description:  the inverse reverse cumulative (upper tail or survivor)
noncentral chi-squared distribution: if
nchi2tail(df,np,x) = p, then invnchi2tail(df,np,p) = x
Domain df:    2e-10 to 1e+6 (may be nonintegral)
Domain np:    0 to 10,000
Domain p:     0 to 1
Range:        0 to 8e+307

npnchi2(df,x,p)
Description:  the noncentrality parameter, np, for the noncentral
chi-squared: if nchi2(df,np,x) = p, then npnchi2(df,x,p)
= np
Domain df:    2e-10 to 1e+6 (may be nonintegral)
Domain x:     0 to 8e+307
Domain p:     0 to 1
Range:        0 to 10,000

Dunnett's multiple range distribution

dunnettprob(k,df,x)
Description:  the cumulative multiple range distribution that is used
in Dunnett's multiple-comparison method with k ranges
and df degrees of freedom; 0 if x < 0

dunnettprob() is computed using an algorithm described
in Miller (1981).
Domain k:     2 to 1e+6
Domain df:    2 to 1e+6
Domain x:     -8e+307 to 8e+307; interesting domain is x > 0
Range:        0 to 1

invdunnettprob(k,df,p)
Description:  the inverse cumulative multiple range distribution that
is used in Dunnett's multiple-comparison method with k
ranges and df degrees of freedom

If dunnettprob(k,df,x) = p, then invdunnettprob(k,df,p)
= x.

invdunnettprob() is computed using an algorithm
described in Miller (1981).
Domain k:     2 to 1e+6
Domain df:    2 to 1e+6
Domain p:     0 to 1 (right exclusive)
Range:        0 to 8e+307

Exponential distribution

exponentialden(b,x)
Description:  the probability density function of the exponential
distribution with scale b

The probability density function of the exponential
distribution is

1/b exp(-x/b)

where b is the scale and x is the value of an
exponential variate.
Domain b:     1e-323 to 8e+307
Domain x:     -8e+307 to 8e+307; interesting domain is x > 0
Range:        1e-323 to 8e+307

exponential(b,x)
Description:  the cumulative exponential distribution with scale b

The cumulative distribution function of the exponential
distribution is

1 - exp(-x/b)

for x > 0 and 0 for x < 0, where b is the scale and x is
the value of an exponential variate.
The mean of the exponential distribution is b and its
variance is b^2.
Domain b:     1e-323 to 8e+307
Domain x:     -8e+307 to 8e+307; interesting domain is x > 0
Range:        0 to 1

exponentialtail(b,x)
Description:  the reverse cumulative exponential distribution with
scale b

The reverse cumulative distribution function of the
exponential distribution is

exp(-x/b)

where b is the scale and x is the value of an
exponential variate.
Domain b:     1e-323 to 8e+307
Domain x:     -8e+307 to 8e+307; interesting domain is x > 0
Range:        0 to 1

invexponential(b,p)
Description:  the inverse cumulative exponential distribution with
scale b: if exponential(b,x) = p, then
invexponential(b,p) = x
Domain b:     1e-323 to 8e+307
Domain p:     0 to 1
Range:        1e-323 to 8e+307

invexponentialtail(b,p)
Description:  the inverse reverse cumulative exponential distribution
with scale b: if exponentialtail(b,x) = p, then
invexponential(b,p) = x
Domain b:     1e-323 to 8e+307
Domain p:     0 to 1
Range:        1e-323 to 8e+307

F and noncentral F distributions

Fden(df1,df2,f)
Description:  the probability density function for the F distribution
with df1 numerator and df2 denominator degrees of
freedom; 0 if f < 0
Domain df1:   1e-323 to 8e+307 (may be nonintegral)
Domain df2:   1e-323 to 8e+307 (may be nonintegral)
Domain f:     -8e+307 to 8e+307; interesting domain is f > 0
Range:        0 to 8e+307

F(df1,df2,f)
Description:  the cumulative F distribution with df1 numerator and df2
denominator degrees of freedom; 0 if f < 0
Domain df1:   2e-10 to 2e+17 (may be nonintegral)
Domain df2:   2e-10 to 2e+17 (may be nonintegral)
Domain f:     -8e+307 to 8e+307; interesting domain is f > 0
Range:        0 to 1

Ftail(df1,df2,f)
Description:  the reverse cumulative (upper tail or survivor) F
distribution with df1 numerator and df2 denominator
degrees of freedom; 1 if f < 0

Ftail(df1,df2,f) = 1 - F(df1,df2,f)
Domain df1:   2e-10 to 2e+17 (may be nonintegral)
Domain df2:   2e-10 to 2e+17 (may be nonintegral)
Domain f:     -8e+307 to 8e+307; interesting domain is f > 0
Range:        0 to 1

invF(df1,df2,p)
Description:  the inverse cumulative F distribution: if F(df1,df2,f) =
p, then invF(df1,df2,p) = f
Domain df1:   2e-10 to 2e+17 (may be nonintegral)
Domain df2:   2e-10 to 2e+17 (may be nonintegral)
Domain p:     0 to 1
Range:        0 to 8e+307

invFtail(df1,df2,p)
Description:  the inverse reverse cumulative (upper tail or survivor)
F distribution: if Ftail(df1,df2,f) = p, then
invFtail(df1,df2,p) = f
Domain df1:   2e-10 to 2e+17 (may be nonintegral)
Domain df2:   2e-10 to 2e+17 (may be nonintegral)
Domain p:     0 to 1
Range:        0 to 8e+307

nFden(df1,df2,np,f)
Description:  the probability density function of the noncentral F
density with df1 numerator and df2 denominator degrees
of freedom and noncentrality parameter np; 0 if f < 0

nFden(df1,df2,0,f) = Fden(df1,df2,f), but Fden() is the
preferred function to use for the central F
distribution.

Also, if F follows the noncentral F distribution with
df1 and df2 degrees of freedom and noncentrality
parameter np, then

df1 F
-----------
df2 + df1 F

follows a noncentral beta distribution with shape
parameters a=df1/2, b=df2/2, and noncentrality parameter
np, as given in nbetaden().  nFden() is computed based
on this relationship.
Domain df1:   1e-323 to 8e+307 (may be nonintegral)
Domain df2:   1e-323 to 8e+307 (may be nonintegral)
Domain np:    0 to 1,000
Domain f:     -8e+307 to 8e+307; interesting domain is f > 0
Range:        0 to 8e+307

nF(df1,df2,np,f)
Description:  the cumulative noncentral F distribution with df1
numerator and df2 denominator degrees of freedom and
noncentrality parameter np; 0 if f < 0

nF(df1,df2,0,f) = F(df1,df2,f)

nF() is computed using nibeta() based on the
relationship between the noncentral beta and noncentral
F distributions:
nF(df1,df2,np,f) =
nibeta(df1/2, df2/2, np, df1*f/{(df1*f)+df2}}).

Domain df1:   2e-10 to 1e+8 (may be nonintegral)
Domain df2:   2e-10 to 1e+8 (may be nonintegral)
Domain np:    0 to 10,000
Domain f:     -8e+307 to 8e+307
Range:        0 to 1

nFtail(df1,df2,np,f)
Description:  the reverse cumulative (upper tail or survivor)
noncentral F distribution with df1 numerator and df2
denominator degrees of freedom and noncentrality
parameter np; 1 if f < 0

nFtail() is computed using nibeta() based on the
relationship between the noncentral beta and F
distributions.  See Johnson, Kotz, and Balakrishnan
(1995) for more details.
Domain df1:   1e-323 to 8e+307 (may be nonintegral)
Domain df2:   1e-323 to 8e+307 (may be nonintegral)
Domain np:    0 to 1,000
Domain f:     -8e+307 to 8e+307; interesting domain is f > 0
Range:        0 to 1

invnF(df1,df2,np,p)
Description:  the inverse cumulative noncentral F distribution: if
nF(df1,df2,np,f) = p, then invnF(df1,df2,np,p) = f
Domain df1:   1e-6 to 1e+6 (may be nonintegral)
Domain df2:   1e-6 to 1e+6 (may be nonintegral)
Domain np:    0 to 10,000
Domain p:     0 to 1
Range:        0 to 8e+307

invnFtail(df1,df2,np,p)
Description:  the inverse reverse cumulative (upper tail or survivor)
noncentral F distribution: if nFtail(df1,df2,np,f) = p,
then invnFtail(df1,df2,np,p) = f
Domain df1:   1e-323 to 8e+307 (may be nonintegral)
Domain df2:   1e-323 to 8e+307 (may be nonintegral)
Domain np:    0 to 1,000
Domain p:     0 to 1
Range:        0 to 8e+307

npnF(df1,df2,f,p)
Description:  the noncentrality parameter, np, for the noncentral F:
if nF(df1,df2,np,f) = p, then npnF(df1,df2,f,p) = np
Domain df1:   2e-10 to 1e+6 (may be nonintegral)
Domain df2:   2e-10 to 1e+6 (may be nonintegral)
Domain f:     0 to 8e+307
Domain p:     0 to 1
Range:        0 to 10,000

Gamma distribution

Description:  the probability density function of the gamma
distribution; 0 if x < g

a is the shape parameter, b is the scale parameter, and
g is the location parameter.
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain g:     -8e+307 to 8e+307
Domain x:     -8e+307 to 8e+307; interesting domain is x > g
Range:        0 to 8e+307

gammap(a,x)
Description:  the cumulative gamma distribution with shape parameter
a; 0 if x < 0

The cumulative Poisson (the probability of observing k
or fewer events if the expected is x) can be evaluated
as 1-gammap(k+1,x). The reverse cumulative (the
probability of observing k or more events) can be
evaluated as gammap(k,x).

gammap() is also known as the incomplete gamma function
(ratio).

Probabilities for the three-parameter gamma distribution
(see gammaden()) can be calculated by shifting and
scaling x; that is, gammap(a,(x - g)/b).
Domain a:     1e-10 to 1e+17
Domain x:     -8e+307 to 8e+307; interesting domain is x > 0
Range:        0 to 1

gammaptail(a,x)
Description:  the reverse cumulative (upper tail or survivor) gamma
distribution with shape parameter a; 1 if x < 0

gammaptail() is also known as the complement to the
incomplete gamma function (ratio).
Domain a:     1e-10 to 1e+17
Domain x:     -8e+307 to 8e+307; interesting domain is x > 0
Range:        0 to 1

invgammap(a,p)
Description:  the inverse cumulative gamma distribution: if
gammap(a,x) = p, then invgammap(a,p) = x
Domain a:     1e-10 to 1e+17
Domain p:     0 to 1
Range:        0 to 8e+307

invgammaptail(a,p)
Description:  the inverse reverse cumulative (upper tail or survivor)
gamma distribution: if gammaptail(a,x) = p, then
invgammaptail(a,p) = x
Domain a:     1e-10 to 1e+17
Domain p:     0 to 1
Range:        0 to 8e+307

dgammapda(a,x)
Description:  the partial derivative of the cumulative gamma
distribution gammap(a,x) with respect to a, for a > 0; 0
if x < 0
Domain a:     1e-7 to 1e+17
Domain x:     -8e+307 to 8e+307; interesting domain is x > 0
Range:        -16 to 0

Description:  the 2nd partial derivative of the cumulative gamma
distribution gammap(a,x) with respect to a, for a > 0; 0
if x < 0
Domain a:     1e-7 to 1e+17
Domain x:     -8e+307 to 8e+307; interesting domain is x > 0
Range:        -0.02 to 4.77e+5

Description:  the 2nd partial derivative of the cumulative gamma
distribution gammap(a,x) with respect to a and x, for a
> 0; 0 if x < 0
Domain a:     1e-7 to 1e+17
Domain x:     -8e+307 to 8e+307; interesting domain is x > 0
Range:        -0.04 to 8e+307

dgammapdx(a,x)
Description:  the partial derivative of the cumulative gamma
distribution gammap(a,x) with respect to x, for a > 0; 0
if x < 0
Domain a:     1e-10 to 1e+17
Domain x:     -8e+307 to 8e+307; interesting domain is x > 0
Range:        0 to 8e+307

dgammapdxdx(a,x)
Description:  the 2nd partial derivative of the cumulative gamma
distribution gammap(a,x) with respect to x, for a > 0; 0
if x < 0
Domain a:     1e-10 to 1e+17
Domain x:     -8e+307 to 8e+307; interesting domain is x > 0
Range:        0 to 1e+40

Description:  the natural logarithm of the inverse gamma density,
where a is the shape parameter and b is the scale
parameter
Domain a:     1e-300 to 1e+300
Domain b:     1e-300 to 1e+300
Domain x:     1e-300 to 8e+307
Range:        -8e+307 to 8e+307

Hypergeometric distribution

hypergeometricp(N,K,n,k)
Description:  the hypergeometric probability of k successes out of a
sample of size n, from a population of size N containing
K elements that have the attribute of interest

Success is obtaining an element with the attribute of
interest.
Domain N:     2 to 1e+5
Domain K:     1 to N-1
Domain n:     1 to N-1
Domain k:     max(0,n-N+K) to min(K,n)
Range:        0 to 1 (right exclusive)

hypergeometric(N,K,n,k)
Description:  the cumulative probability of the hypergeometric
distribution

N is the population size, K is the number of elements in
the population that have the attribute of interest, and
n is the sample size.  Returned is the probability of
observing k or fewer elements from a sample of size n
that have the attribute of interest.
Domain N:     2 to 1e+5
Domain K:     1 to N-1
Domain n:     1 to N-1
Domain k:     max(0,n-N+K) to min(K,n)
Range:        0 to 1

Inverse Gaussian distribution

igaussianden(m,a,x)
Description:  the probability density of the inverse Gaussian
distribution with mean m and shape parameter a; 0 if x <
0
Domain m:     1e-323 to 8e+307
Domain a:     1e-323 to 8e+307
Domain x:     -8e+307 to 8e+307
Range:        0 to 8e+307

igaussian(m,a,x)
Description:  the cumulative inverse Gaussian distribution with mean m
and shape parameter a; 0 if x < 0
Domain m:     1e-323 to 8e+307
Domain a:     1e-323 to 8e+307
Domain x:     -8e+307 to 8e+307
Range:        0 to 1

igaussiantail(m,a,x)
Description:  the reverse cumulative (upper tail or survivor) inverse
Gaussian distribution with mean m and shape parameter a;
1 if x < 0

igaussiantail(m,a,x) = 1 - igaussian(m,a,x)
Domain m:     1e-323 to 8e+307
Domain a:     1e-323 to 8e+307
Domain x:     -8e+307 to 8e+307
Range:        0 to 1

invigaussian(m,a,p)
Description:  the inverse of igaussian(): if igaussian(m,a,x) = p,
then invigaussian(m,a,p) = x
Domain m:     1e-323 to 8e+307
Domain a:     1e-323 to 1e+8
Domain p:     0 to 1 (exclusive)
Range:        0 to 8e+307

invigaussiantail(m,a,p)
Description   the inverse of igaussiantail(): if igaussiantail(m,a,x)
= p, then invigaussiantail(m,a,p) = x
Domain m:     1e-323 to 8e+307
Domain a:     1e-323 to 1e+8
Domain p:     0 to 1 (exclusive)
Range:        0 to 8e+307

lnigaussianden(m,a,x)
Description:  the natural logarithm of the inverse Gaussian density
with mean m and shape parameter a
Domain m:     1e-323 to 8e+307
Domain a:     1e-323 to 8e+307
Domain x:     1e-323 to 8e+307
Range:        -8e+307 to 8e+307

Laplace distribution

laplaceden(m,b,x)
Description:  the probability density of the Laplace distribution with
mean m and scale parameter b
Domain m:     -8e+307 to 8e+307
Domain b:     1e-307 to 8e+307
Domain x:     -8e+307 to 8e+307
Range:        0 to 8e+307

laplace(m,b,x)
Description:  the cumulative Laplace distribution with mean m and
scale parameter b
Domain m:     -8e+307 to 8e+307
Domain b:     1e-307 to 8e+307
Domain x:     -8e+307 to 8e+307
Range:        0 to 1

laplacetail(m,b,x)
Description:  the reverse cumulative (upper tail or survivor) Laplace
distribution with mean m and scale parameter b

laplacetail(m,b,x) = 1 - laplace(m,b,x)
Domain m:     -8e+307 to 8e+307
Domain b:     1e-307 to 8e+307
Domain x:     -8e+307 to 8e+307
Range:        0 to 1

invlaplace(m,b,p)
Description:  the inverse of laplace(): if laplace(m,b,x) = p, then
invlaplace(m,b,p) = x
Domain m:     -8e+307 to 8e+307
Domain b:     1e-307 to 8e+307
Domain p:     0 to 1 (exclusive)
Range:        -8e+307 to 8e+307

invlaplacetail(m,b,p)
Description   the inverse of laplacetail(): if laplacetail(m,b,x) = p,
then invlaplacetail(m,b,p) = x
Domain m:     -8e+307 to 8e+307
Domain b:     1e-307 to 8e+307
Domain p:     0 to 1 (exclusive)
Range:        -8e+307 to 8e+307

lnlaplaceden(m,b,x)
Description:  the natural logarithm of the density of the Laplace
distribution with mean m and scale parameter b
Domain m:     -8e+307 to 8e+307
Domain b:     1e-307 to 8e+307
Domain x:     -8e+307 to 8e+307
Range:        -8e+307 to 707

Logistic distribution

logisticden(x)
Description:  the density of the logistic distribution with mean 0 and
standard deviation pi/sqrt(3)

logisticden(x) = logisticden(1,x) = logisticden(0,1,x),
where x is the value of a logistic random variable.
Domain x:     -8e+307 to 8e+307
Range:        0 to 0.25

logisticden(s,x)
Description:  the density of the logistic distribution with mean 0,
scale s, and standard deviation s pi/sqrt(3)

logisticden(s,x) = logisticden(0,s,x), where s is the
scale and x is the value of a logistic random variable.
Domain s:     1e-323 to 8e+307
Domain x:     -8e+307 to 8e+307
Range:        0 to 8e+307

logisticden(m,s,x)
Description:  the density of the logistic distribution with mean m,
scale s, and standard deviation s pi/sqrt(3)
Domain m:     -8e+307 to 8e+307
Domain s:     1e-323 to 8e+307
Domain x:     -8e+307 to 8e+307
Range:        0 to 8e+307

logistic(x)
Description:  the cumulative logistic distribution with mean 0 and
standard deviation pi/sqrt(3)

logistic(x) = logistic(1,x) = logistic(0,1,x), where x
is the value of a logistic random variable.
Domain x:     -8e+307 to 8e+307
Range:        0 to 1

logistic(s,x)
Description:  the cumulative logistic distribution with mean 0, scale
s, and standard deviation s pi/sqrt(3)

logistic(s,x) = logistic(0,s,x), where s is the scale
and x is the value of a logistic random variable.
Domain s:     1e-323 to 8e+307
Domain x:     -8e+307 to 8e+307
Range:        0 to 1

logistic(m,s,x)
Description:  the cumulative logistic distribution with mean m, scale
s, and standard deviation s pi/sqrt(3)
Domain m:     -8e+307 to 8e+307
Domain s:     1e-323 to 8e+307
Domain x:     -8e+307 to 8e+307
Range:        0 to 1

logistictail(x)
Description:  the reverse cumulative logistic distribution with mean 0
and standard deviation pi/sqrt(3)

logistictail(x) = logistictail(1,x) =
logistictail(0,1,x), where x is the value of a logistic
random variable.
Domain x:     -8e+307 to 8e+307
Range:        0 to 1

logistictail(s,x)
Description:  the reverse cumulative logistic distribution with mean
0, scale s, and standard deviation s pi/sqrt(3)

logistictail(s,x) = logistictail(0,s,x), where s is the
scale and x is the value of a logistic random variable.
Domain s:     1e-323 to 8e+307
Domain x:     -8e+307 to 8e+307
Range:        0 to 1

logistictail(m,s,x)
Description:  the reverse cumulative logistic distribution with mean
m, scale s, and standard deviation s pi/sqrt(3)
Domain m:     -8e+307 to 8e+307
Domain s:     1e-323 to 8e+307
Domain x:     -8e+307 to 8e+307
Range:        0 to 1

invlogistic(p)
Description:  the inverse cumulative logistic distribution: if
logistic(x) = p, then invlogistic(p) = x
Domain p:     0 to 1
Range:        -8e+307 to 8e+307

invlogistic(s,p)
Description:  the inverse cumulative logistic distribution: if
logistic(s,x) = p, then invlogistic(s,p) = x
Domain s:     1e-323 to 8e+307
Domain p:     0 to 1
Range:        -8e+307 to 8e+307

invlogistic(m,s,p)
Description:  the inverse cumulative logistic distribution: if
logistic(m,s,x) = p, then invlogistic(m,s,p) = x
Domain m:     -8e+307 to 8e+307
Domain s:     1e-323 to 8e+307
Domain p:     0 to 1
Range:        -8e+307 to 8e+307

invlogistictail(p)
Description:  the inverse reverse cumulative logistic distribution:
if logistictail(x) = p, then invlogistictail(p) = x
Domain p:     0 to 1
Range:        -8e+307 to 8e+307

invlogistictail(s,p)
Description:  the inverse reverse cumulative logistic distribution:
if logistictail(s,x) = p, then invlogistictail(s,p) = x
Domain s:     1e-323 to 8e+307
Domain p:     0 to 1
Range:        -8e+307 to 8e+307

invlogistictail(m,s,p)
Description:  the inverse reverse cumulative logistic distribution:
if logistictail(m,s,x) = p, then invlogistictail(m,s,p)
= x
Domain m:     -8e+307 to 8e+307
Domain s:     1e-323 to 8e+307
Domain p:     0 to 1
Range:        -8e+307 to 8e+307

Negative binomial distribution

nbinomialp(n,k,p)
Description:  the negative binomial probability

When n is an integer, nbinomialp() returns the
probability of observing exactly floor(k) failures
before the nth success when the probability of a success
on one trial is p.
Domain n:     1e-10 to 1e+6 (can be nonintegral)
Domain k:     0 to 1e+10
Domain p:     0 to 1 (left exclusive)
Range:        0 to 1

nbinomial(n,k,p)
Description:  the cumulative probability of the negative binomial
distribution

n can be nonintegral.  When n is an integer, nbinomial()
returns the probability of observing k or fewer failures
before the nth success, when the probability of a
success on one trial is p.

The negative binomial distribution function is evaluated
using ibeta().
Domain n:     1e-10 to 1e+17 (can be nonintegral)
Domain k:     0 to 2^53-1
Domain p:     0 to 1 (left exclusive)
Range:        0 to 1

nbinomialtail(n,k,p)
Description:  the reverse cumulative probability of the negative
binomial distribution

When n is an integer, nbinomialtail() returns the
probability of observing k or more failures before the
nth success, when the probability of a success on one
trial is p.

The reverse negative binomial distribution function is
evaluated using ibetatail().
Domain n:     1e-10 to 1e+17 (can be nonintegral)
Domain k:     0 to 2^53-1
Domain p:     0 to 1 (left exclusive)
Range:        0 to 1

invnbinomial(n,k,q)
Description:  the value of the negative binomial parameter, p, such
that q = nbinomial(n,k,p)

invnbinomial() is evaluated using invibeta().
Domain n:     1e-10 to 1e+17 (can be nonintegral)
Domain k:     0 to 2^53-1
Domain q:     0 to 1 (exclusive)
Range:        0 to 1

invnbinomialtail(n,k,q)
Description:  the value of the negative binomial parameter, p, such
that q = nbinomialtail(n,k,p)

invnbinomialtail() is evaluated using invibetatail().
Domain n:     1e-10 to 1e+17 (can be nonintegral)
Domain k:     1 to 2^53-1
Domain q:     0 to 1 (exclusive)
Range:        0 to 1 (exclusive)

Normal (Gaussian), binormal, and multivariate normal distributions

normalden(z)
Description:  the standard normal density
Domain:       -8e+307 to 8e+307
Range:        0 to 0.39894 ...

normalden(x,s)
Description:  the normal density with mean 0 and standard deviation s

normalden(x,1) = normalden(x) and
normalden(x,s) = normalden(x/s)/s
Domain x:     -8e+307 to 8e+307
Domain s:     1e-308 to 8e+307
Range:        0 to 8e+307

normalden(x,m,s)
Description:  the normal density with mean m and standard deviation s

normalden(x,0,s) = normalden(x,s) and
normalden(x,m,s) = normalden((x-m)/s)/s
Domain x:     -8e+307 to 8e+307
Domain m:     -8e+307 to 8e+307
Domain s:     1e-308 to 8e+307
Range:        0 to 8e+307

normal(z)
Description:  the cumulative standard normal distribution
Domain:       -8e+307 to 8e+307
Range:        0 to 1

invnormal(p)
Description:  the inverse cumulative standard normal distribution: if
normal(z) = p, then invnormal(p) = z
Domain:       1e-323 to 1 - 2^(-53)
Range:        -38.449394 to 8.2095362

lnnormalden(z)
Description:  the natural logarithm of the standard normal density
Domain:       -1e+154 to 1e+154
Range:        -5e+307 to -0.91893853 = lnnormalden(0)

lnnormalden(x,s)
Description:  the natural logarithm of the normal density with mean 0
and standard deviation s

lnnormalden(x,1) = lnnormalden(x) and
lnnormalden(x,s) = lnnormalden(x/s) - ln(s)
Domain x:     -8e+307 to 8e+307
Domain s:     1e-323 to 8e+307
Range:        -5e+307 to 742.82799

lnnormalden(x,m,s)
Description:  the natural logarithm of the normal density with mean m
and standard deviation s

lnnormalden(x,0,s) = lnnormalden(x,s) and
lnnormalden(x,m,s) = lnnormalden((x-m)/s) - ln(s)
Domain x:     -8e+307 to 8e+307
Domain m:     -8e+307 to 8e+307
Domain s:     1e-323 to 8e+307
Range:        1e-323 to 8e+307

lnnormal(z)
Description:  the natural logarithm of the cumulative standard normal
distribution
Domain:       -1e+99 to 8e+307
Range:        -5e+197 to 0

binormal(h,k,r)
Description:  the joint cumulative distribution of the bivariate
normal with correlation r

Cumulative over (-inf,h] x (-inf,k]
Domain h:     -8e+307 to 8e+307
Domain k:     -8e+307 to 8e+307
Domain r:     -1 to 1
Range:        0 to 1

lnmvnormalden(M,V,X)
Description:  the natural logarithm of the multivariate normal density

M is the mean vector, V is the covariance matrix, and X
is the random vector.
Domain M:     1 x n and n x 1 vectors
Domain V:     n x n, positive-definite, symmetric matrices
Domain X:     1 x n and n x 1 vectors
Range:        -8e+307 to 8e+307

Poisson distribution

poissonp(m,k)
Description:  the probability of observing floor(k) outcomes that are
distributed as Poisson with mean m

The Poisson probability function is evaluated using
Domain m:     1e-10 to 1e+8
Domain k:     0 to 1e+9
Range:        0 to 1

poisson(m,k)
Description:  the probability of observing floor(k) or fewer outcomes
that are distributed as Poisson with mean m

The Poisson distribution function is evaluated using
gammaptail().
Domain m:     1e-10 to 2^53-1
Domain k:     0 to 2^53-1
Range:        0 to 1

poissontail(m,k)
Description:  the probability of observing floor(k) or more outcomes
that are distributed as Poisson with mean m

The reverse cumulative Poisson distribution function is
evaluated using gammap().
Domain m:     1e-10 to 2^53-1
Domain k:     0 to 2^53-1
Range:        0 to 1

invpoisson(k,p)
Description:  the Poisson mean such that the cumulative Poisson
distribution evaluated at k is p: if poisson(m,k) = p,
then invpoisson(k,p) = m

The inverse Poisson distribution function is evaluated
using invgammaptail().
Domain k:     0 to 2^53-1
Domain p:     0 to 1 (exclusive)
Range:        1.110e-16 to 2^53

invpoissontail(k,q)
Description:  the Poisson mean such that the reverse cumulative
Poisson distribution evaluated at k is q: if
poissontail(m,k) = q, then invpoissontail(k,q) = m

The inverse of the reverse cumulative Poisson
distribution function is evaluated using invgammap().
Domain k:     0 to 2^53-1
Domain q:     0 to 1 (exclusive)
Range:        0 to 2^53 (left exclusive)

Student's t and noncentral Student's t distributions

tden(df,t)
Description:  the probability density function of Student's t
distribution
Domain df:    1e-323 to 8e+307 (may be nonintegral)
Domain t:     -8e+307 to 8e+307
Range:        0 to 0.39894 ...

t(df,t)
Description:  the cumulative Student's t distribution with df degrees
of freedom
Domain df:    2e-10 to 2e+17 (may be nonintegral)
Domain t:     -8e+307 to 8e+307
Range:        0 to 1

ttail(df,t)
Description:  the reverse cumulative (upper tail or survivor)
Student's t distribution; the probability T > t
Domain df:    2e-10 to 2e+17 (may be nonintegral)
Domain t:     -8e+307 to 8e+307
Range:        0 to 1

invt(df,p)
Description:  the inverse cumulative Student's t distribution: if
t(df,t) = p, then invt(df,p) = t
Domain df:    2e-10 to 2e+17 (may be nonintegral)
Domain p:     0 to 1
Range:        -8e+307 to 8e+307

invttail(df,p)
Description:  the inverse reverse cumulative (upper tail or survivor)
Student's t distribution: if ttail(df,t) = p, then
invttail(df,p) = t
Domain df:    2e-10 to 2e+17 (may be nonintegral)
Domain p:     0 to 1
Range:        -8e+307 to 8e+307

invnt(df,np,p)
Description:  the inverse cumulative noncentral Student's t
distribution:  if nt(df,np,t) = p, then invnt(df,np,p) =
t
Domain df:    1 to 1e+6 (may be nonintegral)
Domain np:    -1,000 to 1,000
Domain p:     0 to 1
Range:        -8e+307 to 8e+307

invnttail(df,np,p)
Description:  the inverse reverse cumulative (upper tail or survivor)
noncentral Student's t distribution: if nttail(df,np,t)
= p, then invnttail(df,np,p) = t
Domain df:    1 to 1e+6 (may be nonintegral)
Domain np:    -1,000 to 1,000
Domain p:     0 to 1
Range:        -8e+10 to 8e+10

ntden(df,np,t)
Description:  the probability density function of the noncentral
Student's t distribution with df degrees of freedom and
noncentrality parameter np
Domain df:    1e-100 to 1e+10 (may be nonintegral)
Domain np:    -1,000 to 1,000
Domain t:     -8e+307 to 8e+307
Range:        0 to 0.39894 ...

nt(df,np,t)
Description:  the cumulative noncentral Student's t distribution with
df degrees of freedom and noncentrality parameter np

nt(df,0,t) = t(df,t)
Domain df:    1e-100 to 1e+10 (may be nonintegral)
Domain np:    -1,000 to 1,000
Domain t:     -8e+307 to 8e+307
Range:        0 to 1

nttail(df,np,t)
Description:  the reverse cumulative (upper tail or survivor)
noncentral Student's t distribution with df degrees of
freedom and noncentrality parameter np
Domain df:    1e-100 to 1e+10 (may be nonintegral)
Domain np:    -1,000 to 1,000
Domain t:     -8e+307 to 8e+307
Range:        0 to 1

npnt(df,t,p)
Description:  the noncentrality parameter, np, for the noncentral
Student's t distribution:  if nt(df,np,t) = p, then
npnt(df,t,p) = np
Domain df:    1e-100 to 1e+8 (may be nonintegral)
Domain t:     -8e+307 to 8e+307
Domain p:     0 to 1
Range:        -1,000 to 1,000

Tukey's Studentized range distribution

tukeyprob(k,df,x)
Description:  the cumulative Tukey's Studentized range distribution
with k ranges and df degrees of freedom; 0 if x < 0

If df is a missing value, then the normal distribution
is used instead of Student's t.

tukeyprob() is computed using an algorithm described in
Miller (1981).
Domain k:     2 to 1e+6
Domain df:    2 to 1e+6
Domain x:     -8e+307 to 8e+307
Range:        0 to 1

invtukeyprob(k,df,p)
Description:  the inverse cumulative Tukey's Studentized range
distribution with k ranges and df degrees of freedom

If df is a missing value, then the normal distribution
is used instead of Student's t.  If tukeyprob(k,df,x) =
p, then invtukeyprob(k,df,p) = x.

invtukeyprob() is computed using an algorithm described
in Miller (1981).
Domain k:     2 to 1e+6
Domain df:    2 to 1e+6
Domain p:     0 to 1
Range:        0 to 8e+307

Weibull distribution

weibullden(a,b,x)
Description:  the probability density function of the Weibull
distribution with shape a and scale b

weibullden(a,b,x) = weibullden(a,b,0,x), where a is the
shape, b is the scale, and x is the value of the Weibull
random variable.
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain x:     1e-323 to 8e+307
Range:        0 to 8e+307

weibullden(a,b,g,x)
Description:  the probability density function of the Weibull
distribution with shape a, scale b, and location g
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain g:     -8e+307 to 8e+307
Domain x:     -8e+307 to 8e+307; interesting domain is x > g
Range:        0 to 8e+307

weibull(a,b,x)
Description:  the cumulative Weibull distribution with shape a and
scale b

weibull(a,b,x) = weibull(a,b,0,x), where a is the shape,
b is the scale, and x is the value of the Weibull random
variable.
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain x:     1e-323 to 8e+307
Range:        0 to 1

weibull(a,b,g,x)
Description:  the cumulative Weibull distribution with shape a, scale
b, and location g
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain g:     -8e+307 to 8e+307
Domain x:     -8e+307 to 8e+307; interesting domain is x > g
Range:        0 to 1

weibulltail(a,b,x)
Description:  the reverse cumulative Weibull distribution with shape a
and scale b

weibulltail(a,b,x) = weibulltail(a,b,0,x), where a is
the shape, b is the scale, and x is the value of a
Weibull random variable.
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain x:     1e-323 to 8e+307
Range:        0 to 1

weibulltail(a,b,g,x)
Description:  the reverse cumulative Weibull distribution with shape
a, scale b, and location g
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain g:     -8e+307 to 8e+307
Domain x:     -8e+307 to 8e+307; interesting domain is x > g
Range:        0 to 1

invweibull(a,b,p)
Description:  the inverse cumulative Weibull distribution with shape a
and scale b: if weibull(a,b,x) = p, then
invweibull(a,b,p) = x
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain p:     0 to 1
Range:        1e-323 to 8e+307

invweibull(a,b,g,p)
Description:  the inverse cumulative Weibull distribution with shape
a, scale b, and location g: if weibull(a,b,g,x) = p,
then invweibull(a,b,g,p) = x
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain g:     -8e+307 to 8e+307
Domain p:     0 to 1
Range:        g+c(epsdouble) to 8e+307

invweibulltail(a,b,p)
Description:  the inverse reverse cumulative Weibull distribution with
shape a and scale b: if weibulltail(a,b,x) = p, then
invweibulltail(a,b,p) = x
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain p:     0 to 1
Range:        1e-323 to 8e+307

invweibulltail(a,b,g,p)
Description:  the inverse reverse cumulative Weibull distribution with
shape a, scale b, and location g: if
weibulltail(a,b,g,x) = p, then invweibulltail(a,b,g,p) =
x
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain g:     -8e+307 to 8e+307
Domain p:     0 to 1
Range:        g+c(epsdouble) to 8e+307

Weibull (proportional hazards) distribution

weibullphden(a,b,x)
Description:  the probability density function of the Weibull
(proportional hazards) distribution with shape a and
scale b

weibullphden(a,b,x) = weibullphden(a,b,0,x), where a is
the shape, b is the scale, and x is the value of the
Weibull (proportional hazards) random variable.
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain x:     1e-323 to 8e+307
Range:        0 to 8e+307

weibullphden(a,b,g,x)
Description:  the probability density function of the Weibull
(proportional hazards) distribution with shape a, scale
b, and location g
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain g:     -8e+307 to 8e+307
Domain x:     -8e+307 to 8e+307; interesting domain is x > g
Range:        0 to 8e+307

weibullph(a,b,x)
Description:  the cumulative Weibull (proportional hazards)
distribution with shape a and scale b

weibullph(a,b,x) = weibullph(a,b,0,x), where a is the
shape, b is the scale, and x is the value of the Weibull
random variable.
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain x:     1e-323 to 8e+307
Range:        0 to 1

weibullph(a,b,g,x)
Description:  the cumulative Weibull (proportional hazards)
distribution with shape a, scale b, and location g
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain g:     -8e+307 to 8e+307
Domain x:     -8e+307 to 8e+307; interesting domain is x > g
Range:        0 to 1

weibullphtail(a,b,x)
Description:  the reverse cumulative Weibull (proportional hazards)
distribution with shape a and scale b

weibullphtail(a,b,x) = weibullphtail(a,b,0,x), where a
is the shape, b is the scale, and x is the value of the
Weibull (proportional hazards) random variable.
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain x:     1e-323 to 8e+307
Range:        0 to 1

weibullphtail(a,b,g,x)
Description:  the reverse cumulative Weibull (proportional hazards)
distribution with shape a, scale b, and location g
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain g:     -8e+307 to 8e+307
Domain x:     -8e+307 to 8e+307; interesting domain is x > g
Range:        0 to 1

invweibullph(a,b,p)
Description:  the inverse cumulative Weibull (proportional hazards)
distribution with shape a and scale b: if
weibullph(a,b,x) = p, then invweibullph(a,b,p) = x
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain p:     0 to 1
Range:        1e-323 to 8e+307

invweibullph(a,b,g,p)
Description:  the inverse cumulative Weibull (proportional hazards)
distribution with shape a, scale b, and location g: if
weibullph(a,b,g,x) = p, then invweibullph(a,b,g,p) = x
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain g:     -8e+307 to 8e+307
Domain p:     0 to 1
Range:        g+c(epsdouble) to 8e+307

invweibullphtail(a,b,p)
Description:  the inverse reverse cumulative Weibull (proportional
hazards) distribution with shape a and scale b: if
weibullphtail(a,b,x) = p, then invweibullphtail(a,b,p) =
x
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain p:     0 to 1
Range:        1e-323 to 8e+307

invweibullphtail(a,b,g,p)
Description:  the inverse reverse cumulative Weibull (proportional
hazards) distribution with shape a, scale b, and
location g: if weibullphtail(a,b,g,x) = p, then
invweibullphtail(a,b,g,p) = x
Domain a:     1e-323 to 8e+307
Domain b:     1e-323 to 8e+307
Domain g:     -8e+307 to 8e+307
Domain p:     0 to 1
Range:        g+c(epsdouble) to 8e+307

Wishart distribution

lnwishartden(df,V,X)
Description:  the natural logarithm of the density of the Wishart
distribution; missing if df <= n-1

df denotes the degrees of freedom, V is the scale
matrix, and X is the Wishart random matrix.
Domain df:    1 to 1e+100 (may be nonintegral)
Domain V:     n x n, positive-definite, symmetric matrices
Domain X:     n x n, positive-definite, symmetric matrices
Range:        -8e+307 to 8e+307

lniwishartden(df,V,X)
Description:  the natural logarithm of the density of the inverse
Wishart distribution; missing if df <= n-1

df denotes the degrees of freedom, V is the scale
matrix, and X is the inverse Wishart random matrix.
Domain df:    1 to 1e+100 (may be nonintegral)
Domain V:     n x n, positive-definite, symmetric matrices
Domain X:     n x n, positive-definite, symmetric matrices
Range:        -8e+307 to 8e+307

References

Johnson, N. L., S. Kotz, and N. Balakrishnan.  1995.  Continuous
Univariate Distributions, Vol. 2.  2nd ed.  New York: Wiley.

Miller, R. G., Jr.  1981.  Simultaneous Statistical Inference.  2nd ed.
New York: Springer.

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