Stata 15 help for density_functions

[FN] Statistical functions

Functions

The probability distribution and density functions are organized under the following headings:

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

betaden(a,b,x) 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

nbetaden(a,b,np,x) 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.

nbetaden(a,b,0,x) = betaden(a,b,x), but betaden() is the 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 chi2den(df,x) = gammaden(df/2,2,0,x) 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

gammaden(a,b,g,x) 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

dgammapdada(a,x) 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

dgammapdadx(a,x) 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

lnigammaden(a,b,x) 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 gammaden(). 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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